Tame algebras and Tits quadratic forms

Advances in Mathematics 226 (2011) 887–951www.elsevier.com/locate/aim

Tame algebras and Tits quadratic forms

Thomas Brüstle a,b,1, José Antonio de la Peña c,2, Andrzej Skowronski d,∗,3

a Département de Mathématiques et Informatique, Université de Sherbrooke, Sherbrooke, Québec, J1K 2R1 Canadab Division of Natural Sciences and Mathematics, Bishop’s University, Lennoxville, Québec, J1M 1Z7 Canada

c Instituto de Mathematicas, UNAM, Ciudad Universitaria, México 04510, D. F., Mexicod Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, PL-87-100 Torun,

Poland

Received 25 June 2009; accepted 24 July 2010

Available online 5 August 2010

Communicated by Henning Krause

Abstract

We solve a long standing problem concerning the connection between the tameness of simply connectedalgebras and the weak nonnegativity of the associated Tits integral quadratic forms, and derive some con-sequences.© 2010 Elsevier Inc. All rights reserved.

MSC: 16G20; 16G70

Keywords: Tame algebras; Tits forms; Simply connected algebras; Degenerations of algebras

0. Introduction and the main results

Throughout the paper, K will denote a fixed algebraically closed field. By an algebra A ismeant an associative, finite dimensional K-algebra with an identity, which we shall assume(without loss of generality) to be basic and connected. Such an algebra A has a presentationA ∼= KQ/I , where KQ is the path algebra of the Gabriel quiver Q = QA of A and I is an ad-

* Corresponding author.E-mail addresses: [email protected] (T. Brüstle), [email protected] (J.A. de la Peña),

[email protected] (A. Skowronski).1 Supported by the NSERC of Canada.2 Supported by the Conacyt Grant of México.3 Supported by the Polish Scientific Grant KBN No. 1 P03A 018 27.

0001-8708/$ – see front matter © 2010 Elsevier Inc. All rights reserved.doi:10.1016/j.aim.2010.07.007

888 T. Brüstle et al. / Advances in Mathematics 226 (2011) 887–951

missible ideal in KQ. Equivalently, an algebra A = KQ/I may be considered as a K-categorywhose class of objects is the set of vertices of Q, and the space of morphisms A(x,y) from x

to y is the quotient of the K-space KQ(x,y) of K-linear combinations of paths in Q from x to y

by the subspace I (x, y) = KQ(x,y)∩ I . An algebra A with QA having no oriented cycle is saidto be triangular. A full subcategory C of A is said to be convex if any path in QA with sourceand sink in QC lies entirely in QC . For an algebra A, we denote by modA the category of finitedimensional right A-modules and by indA the full subcategory consisting of indecomposablemodules. The term A-module is used for an object of modA if not specified otherwise.

From Drozd’s Tame and Wild Theorem [30] (see also [21,32]) the algebras may be dividedinto two disjoint classes. One class consists of the tame algebras for which the indecompos-able modules occur, in each dimension d , in a finite number of discrete and a finite numberof one-parameter families. The second class is formed by the wild algebras whose representa-tion theory comprises the representation theories of all algebras. Hence, we may realisticallyhope to classify the indecomposable modules only for the tame algebras. More precisely, fol-lowing [30], an algebra A is called tame if, for each dimension d , there exists a finite number ofK[X]-A-bimodules Mi which are finitely generated and free as left K[X]-modules, and all buta finite number of isomorphism classes of indecomposable A-modules of dimension d are of theform K[X]/(X − λ) ⊗K[X] Mi for some i and some λ ∈ K . Among the tame algebras we maydistinguish the class of representation-finite algebras, having only finitely many isomorphismclasses of indecomposable modules, for which the representation theory is presently rather wellunderstood (see [7,12,13,17]), and may be reduced (via coverings) to the representation theoryof representation-finite (strongly) simply connected algebras. On the other hand, the representa-tion theory of arbitrary tame algebras is still only emerging. Frequently, applying deformationsand covering techniques, we may reduce the study of modules over tame algebras to that for thecorresponding simply connected algebras. Here, we are concerned with the problem of findingcombinatorial criteria for a simply connected algebra to be tame.

In the fundamental paper [31] from 1972, P. Gabriel has proved that the path algebra KQ of afinite connected quiver Q is representation-finite if and only if the associated Tits quadratic formof Q is positive. One year later, L.A. Nazarova [43] (see also [22,23]) proved that the path algebraKQ of a finite connected quiver Q is tame if and only if the Tits form of Q is nonnegative.In 1975 S. Brenner [16] initiated the study of connections between the representation type ofalgebras given by quivers with relations and the definiteness of certain quadratic forms, andwrote: “This paper is written in the spirit of experimental science. It reports some observedregularities and suggests that there should be a theory to explain them”. In 1983 K. Bongartz [10]associated a Tits quadratic form to any triangular algebra. The Tits form of a triangular algebraA = KQ/I is an integral quadratic form qA : ZQ0 → Z, defined, for x = (xi) ∈ Z

Q0 , by

qA(x) =∑i∈Q0

x2i −

∑(i→j)∈Q1

xixj +∑

i,j∈Q0

r(i, j)xixj ,

where Q0 is the set of vertices of Q, Q1 the set of arrows of Q, and r(i, j) = |R ∩ I (i, j)|,for a minimal set of generators R ⊂ ⋃

i,j∈Q0I (i, j) of the admissible ideal I . It follows from

Krull’s Principal Ideal Theorem that qA(d) � dimG(d) − dim modA(d) for any d ∈ NQ0 ,

where modA(d) is the affine variety of A-modules of dimension-vector d and G(d) is theproduct

∏i∈Q0

GLdi(K) of general linear groups acting on modA(d) by conjugations. It

has been observed in [10], generalizing the Tits observation, (respectively, in [49]) that ifA is representation-finite (respectively, tame) then dimG(d) > dim modA(d) (respectively,

T. Brüstle et al. / Advances in Mathematics 226 (2011) 887–951 889

dimG(d) � dim modA(d)), and consequently qA is weakly positive (positive on positive vectors)(respectively, weakly nonnegative (nonnegative on nonnegative vectors)). The reverse implica-tions have been proved for some classes of algebras with small homological dimensions (tiltedalgebras [36], double tilted algebras [61], quasitilted algebras [72], coil enlargements of tameconcealed algebras [5], algebras with separating almost cyclic coherent Auslander–Reiten com-ponents [40], one-point extensions of tame concealed algebras [47], multiple regular extensionsof tame concealed algebras [29]). Unfortunately, these implications are not true for arbitrarytriangular algebras: there are wild triangular algebras (even of global dimension 2) with weaklypositive Tits form (see for example [10]). One has to impose some nondegeneracy conditions ona triangular algebra A to recover its representation type from the weak positivity (respectively,weak nonnegativity) of the Tits form qA. A natural and important condition is the simple con-nectedness of an algebra (see Section 1 for the relevant definitions). In [10] K. Bongartz provedthat if the Auslander–Reiten quiver of a triangular algebra A admits a preprojective componentthen A is representation-finite if and only if the Tits form qA of A is weakly positive. In par-ticular, this implies that a simply connected algebra A is representation-finite if and only if theTits form qA of A is weakly positive. Unfortunately, this cannot be extended to the tame alge-bras, because there are (see Section 1) wild simply connected algebras (even with preprojectivecomponent in the Auslander–Reiten quiver) having weakly nonnegative Tits form. But everyrepresentation-finite simply connected algebra A is strongly simply connected [69], that is, everyfull convex subcategory C of A is simply connected.

The following main result of the paper is a natural extension of the Bongartz result to the tamealgebras, and solves the problem raised by S. Brenner more than 30 years ago.

Main Theorem. Let A be a strongly simply connected algebra. Then A is tame if and only if theTits form qA of A is weakly nonnegative.

For the (very) special class of strongly simply connected algebras formed by the tree algebras(the Gabriel quiver is a tree) this fact has been proved by the first named author in [19]. We alsonote (see [48,77]) that a unit integral quadratic form q : Zn → Z is weakly nonnegative if andonly if q(z) � 0 for every z ∈ [0,12]n. We point out that this provides an easy combinatorialcriterion to check the tameness of strongly simply connected algebras.

It is known [12] that a strongly simply connected algebra A is representation-finite if and onlyif A does not contain a convex subcategory (called a critical algebra) which is a preprojectivetilt of a hereditary algebra of an Euclidean type Dn, E6, E7, or E8. Moreover, it follows froma result by J.A. de la Peña [48] that the Tits form of a strongly simply connected algebra A isweakly nonnegative if and only if A does not contain a convex subcategory (called a hypercriticalalgebra) which is a preprojective tilt of a wild hereditary algebra of one of tree types

T5 • •• •

• •

˜Dn • •

• • · · · • •• • •˜

E6 ••

• • • • • •

˜E7 •

• • • • • • • •


˜E8 •

• • • • • • • • •

where in the case of ˜Dn the number of vertices is n + 2, 4 � n � 8. The critical (respectively,

hypercritical) algebras have been classified completely by quivers and relations in [11,34] (re-spectively, [39,74,76]). Therefore, we obtain the following consequence of our main theoremwhich gives another handy criterion for a strongly simply connected algebra to be tame.

Corollary 1. Let A be a strongly simply connected algebra. Then A is tame if and only if A doesnot contain a convex hypercritical subcategory.

Since the Gabriel quivers of hypercritical algebras have at most 10 vertices, we obtain alsothe following consequence.

Corollary 2. Let A be a strongly simply connected algebra. Then A is tame if and only if everyconvex subcategory of A with at most 10 objects is tame.

The above corollary has been applied recently by S. Kasjan [35] to show that the class of tamestrongly simply connected algebras forms an open Z-scheme in every dimension. In particular,for any dimension d , the set of points of the affine variety algd(K) of associative K-algebras ofdimension d corresponding to the tame strongly simply connected algebras is an open set in theZariski topology.

Recall that an algebra is called strictly wild if there is a full exact embedding functormodK〈x, y〉 → modA, where K〈x, y〉 is the free (noncommutative) algebra in two variables.It is known [15] that, if A is strictly wild, then for any algebra Λ there is a full exact embeddingmodΛ → modA. Since all hypercritical algebras are strictly wild [36], we obtain the strongerversion of Drozd’s Tame and Wild Theorem for the strongly simply connected algebras.

Corollary 3. Every wild strongly simply connected algebra is strictly wild.

In the representation theory of algebras an essential role is played by the linear represen-tations of partially ordered sets and vector space categories (see [62,63,65] for general theoryand applications). The representation-finite (respectively, tame) partially ordered sets have beencharacterized in 1972 by M. Kleiner [37] (respectively, in 1975 by L.A. Nazarova [44]). It wasshown already by P. Gabriel [31] that the representation theory of representation-finite quiverscan be reduced to that for the representation-finite partially ordered sets. In [14] K. Bongartz andC.M. Ringel proved that a tree algebra A = KQ/I is representation-finite if and only if certainpartially ordered sets Si associated to the vertices i of Q are representation-finite. This raised theproblem of extending the above connection to wider classes of representation-finite (respectively,tame) simply connected algebras. A wide class of simply connected algebras is formed by thecompletely separating algebras [27] (equivalently, schurian strongly simply connected algebras).For a completely separating algebra A = KQ/I and a vertex i of Q one associates (see [28] fordetails) the partially ordered set P (A, i) formed by the isomorphism classes of indecomposablethin start modules. It has been proved in [25] and [28] that a completely separating algebra A isrepresentation-finite if and only if the partially ordered sets P (A, i) associated to all vertices i

of the Gabriel quiver QA of A are representation-finite. Moreover, P. Dräxler and R. Nörenberg


proved in [28] that the Tits form qA of a completely separating algebra A is weakly nonnega-tive if and only if the partially ordered sets P (A, i) are tame. Hence, applying our main resultand the Nazarova’s criterion [44], we obtain the following characterization of tame completelyseparating algebras.

Corollary 4. A completely separating algebra A is tame if and only if the partially orderedsets P (A, i) associated to the vertices i of the Gabriel quiver QA of A do not contain partiallyordered subsets whose Hasse diagrams are of the forms

N1 = (1,1,1,1,1) = • • • • • ,

•

N2 = (1,1,1,2) = • • • • ,

•• • •

N3 = (2,2,3) = • • • ,

•• •• •

N4 = (1,3,4) = • • • ,

•••

• • •

N5 = (N,5) = • • • ,

••••

• •

N6 = (1,2,6) = • • • .

We present now an idea of the proof of our main result. An important class of tame algebrasis formed by the algebras of polynomial growth [68], for which there is a natural number m

such that the number of one-parameter families of indecomposable modules is bounded, in eachdimension d , by dm. The representation theory of strongly simply connected algebras of poly-nomial growth is presently well understood (see [70]), and the prominent role in this theory isplayed by the coil and multicoil algebras introduced by I. Assem and A. Skowronski in [3,4]. Werefer also to [52–54,57] for geometric and homological characterizations as well as properties ofthe Tits forms of strongly simply connected algebras of polynomial growth. Moreover, the classof minimal non-polynomial growth tame strongly simply connected algebras (pg-critical alge-bras) has been described completely by R. Nörenberg and A. Skowronski [46] (see also [45]).The basic ingredient of our proof is the following criterion established by A. Skowronski in [70]:a strongly simply connected algebra A is of polynomial growth if and only if the Tits form qA

of A is weakly nonnegative and A does not contain a pg-critical convex subcategory. Hence,in order to prove that a strongly simply connected algebra with weakly nonnegative Tits formis tame, we may restrict to the algebras A containing a convex pg-critical subcategory, andan indecomposable module whose support contains all sinks and sources of the Gabriel quiver


of A. We prove, applying the main result of [58], that every such an algebra A is a D-algebra(the new concept defined in Section 4, generalizing the concept introduced in [19] for the treealgebras), which is a suitable pushout glueing of blowups of D-coil algebras and pg-critical al-gebras. Further, we show that the representation theory of a D-algebra A is controlled by anotherD-algebra A∗ (canonically associated to A), which degenerates to a special biserial algebra A.Since the special biserial algebras are tame (see [24,20,75]), applying the Geiss degenerationtheorem [33], we conclude that A is tame.

In the course of the proof of our main theorem we obtain also the following characterization ofthe tame strongly connected algebras in terms of the supports of their indecomposable modules(see Proposition 3.2 and Theorem 6.1).

Corollary 5. A strongly simply connected algebra A is tame if and only if the convex hull of thesupport of any indecomposable A-module inside A is an algebra of one of the forms: a tametilted algebra, a coil algebra, or a D-algebra.

We end this section with some comments and open problems concerning the Tits forms and in-decomposable modules over tame strongly simply connected algebras. It follows from [10] that,for a representation-finite (strongly) simply connected algebra A, the dimension vector func-tion induces a bijection between the isomorphism classes of indecomposable A-modules and thepositive roots of the Tits form qA. Moreover, the indecomposable modules over representation-finite simply connected algebras are directing modules and hence their support algebras are tiltedalgebras. These algebras have been classified completely by quivers and relations: they are 24 in-finite regular families of K. Bongartz [9] (see also [63, (6.3)]) whose Gabriel quivers have at least14 vertices, being of considerable theoretical interest (see [24,71,59]), and 16.344 exceptional al-gebras described in [26,64]. For the representation-infinite strongly simply connected algebras,the situation is much more complicated and has to be clarified. In [57, (5.4)] J.A. de la Peñaand A. Skowronski constructed, for any positive integers m, r , a tame (1-parametric) stronglysimply connected algebra A = A(m, r) such that for any n ∈ {1, . . . , r} there are pairwise noni-somorphic indecomposable A-modules X

(n)1 , . . . ,X

(n)m with common dimension vector v(n) and

qA(v(n)) = n. On the other hand, for a fixed strongly simply connected algebra A of polynomialgrowth, there is a common bound (depending only of the number of vertices of the Gabriel quiverof A) on the values of the Tits form qA on the dimension vectors of indecomposable A-modules(see [57, Theorem]). It is also known (see [57, (5.6)]) that there exist tame strongly simply con-nected algebras A without common bound on the values of the Tits form qA on the dimensionvectors of indecomposable A-modules. Further, surprisingly, T. Brüstle exhibited in [18] a min-imal non-polynomial growth tame strongly simply connected (pg-critical) algebra A such thatthe values of its Tits form qA on the dimension vectors of all indecomposable A-modules arebounded by 2. The support algebras of indecomposable directing modules over representation-infinite tame algebras have been investigated by J.A. de la Peña in [50,51]. In particular, it wasshown in [50] that these algebras are at most 2-parametric (have at most 2 one-parameter fami-lies of indecomposable modules of any given dimension). The 2-parametric support algebras ofdirecting indecomposable modules over tame strongly simply connected algebras, with at least20 vertices in the Gabriel quivers, have been classified by quivers and relations in [51]: thereare 19 infinite regular families of such algebras. The classification of the remaining support al-gebras of directing indecomposable modules over representation-infinite tame strongly simplyconnected algebras is still an open problem. On the other hand, a description of the supportalgebras of nondirecting indecomposable modules over strongly simply connected algebras of


polynomial growth follows from the paper by P. Malicki, A. Skowronski and B. Tomé [41], andthe main results of [70]. Finally, we point out that the classification of all nondirecting inde-composable (finite dimensional) modules over arbitrary non-polynomial growth tame stronglysimply connected algebras seems to be a difficult but exciting open problem. In particular, wewould like to describe the dimension vectors of indecomposable modules over an arbitrary tamestrongly simply connected algebra A and the values of the Tits form qA on them.

For basic background on the representation theory applied here we refer to [1,6,38,63,65–67].The main results of the paper have been presented by the authors during conferences and

seminars in Bern, Bielefeld, Boston, Budapest, Oslo, Pátzcuaro, Tokyo, Torun, Trieste.

1. Simply connected algebras

Let A be a triangular algebra and Q its Gabriel quiver. For each vertex x of Q, denote by Q(x)

the subquiver of Q obtained by deleting all those vertices of Q being a source of a path in Q withtarget x (including the trivial path from x to x). We shall denote by A(x) the full subcategoryof A whose objects are the vertices of Q(x). Moreover, for each vertex x of Q, denote by P(x)

the indecomposable projective A-module at x, and by R(x) the radical of P(x). Then R(x) issaid to be separated if R(x) is a direct sum of pairwise nonisomorphic indecomposable moduleswhose supports are contained in pairwise different connected components of Q(x). We say thatA has the separation property [8] if R(x) is separated for any vertex x of Q. It was shown in [69,Proposition 2.3] that if A has the separation property then A is simply connected in the senseof [2], that is, for any presentation A ∼= KQ/I of A as a bound quiver algebra, the fundamentalgroup Π1(Q, I) of (Q, I) is trivial. Recall also that A is called strongly simply connected [69] ifevery convex subcategory of A is simply connected. The following characterization of stronglysimply connected algebras has been established in [69, Theorem 4.1].

Proposition 1.1. For a triangular algebra A the following conditions are equivalent:

(i) A is strongly simply connected.(ii) Every convex subcategory of A has the separation property.

(iii) Every convex subcategory of Aop has the separation property.(iv) The first Hochschild cohomology space H 1(C,C) of any convex subcategory C of A van-

ishes.

The one-point extension of an algebra A by an A-module X is the matrix algebra

A[X] =[

K X

0 A

]with the usual addition and multiplication of matrices. The quiver QA[X] of A[X] contains thequiver QA of A as a convex subquiver and there is an additional (extension) vertex whichis a source. The A[X]-modules are usually identified with triples (V ,M,ϕ), where V is aK-vector space, M is an A-module and ϕ :V → HomA(X,M) is a K-linear map. An A[X]-homomorphism (V ,M,ϕ) → (V ′,M ′, ϕ′) is then a pair (f, g), where f :V → V ′ is a K-homomorphism and g :M → M ′ is an A-homomorphism such that ϕ′f = HomA(X,g)ϕ. Onedefines dually the one-point coextension [X]A of A by X.

We will need also the following fact proved in [55, Proposition 2.2].


Proposition 1.2. Let B be a convex subcategory of a strongly simply connected algebra A. Thenthere is a sequence B = Λ0,Λ1, . . . ,Λm = A of convex subcategories of A such that, for eachi ∈ {0, . . . ,m − 1}, Λi+1 is a one-point extension or coextension of Λi by an indecomposableΛi -module.

The following direct consequence of the proof of [48, Theorem 3.1] will be essential for ourconsiderations.

Proposition 1.3. Let A be a strongly simply connected algebra. Then the Tits form qA of A isweakly nonnegative if and only if A does not contain a convex hypercritical subcategory.

Following [46, (3.2)] by a pg-critical algebra we mean here a bound quiver algebra obtainedfrom one of the frames (1)–(16) below by operations of the following forms:

(a) Replacing each subgraph

•

•

•by

•

•

•

or

•.

..

.•

•(b) Choosing arbitrary orientations in nonoriented edges.(c) Constructing the opposite algebra.

(1) •• · · · •

• •• · · · •

• •• · · · •

•

(2) • • •• · · · • •

• • · · · ••


(3) •• .

.

.• · · · • •

• • • · · · •• •...

•

(4) •• .

..• •• • · · · •• •...

•

(5) •• · · · •

• • •• · · · •

• •

(6) • •• •

• · · · •• •

(7) •• · · · •

• • • •• · · · •

•

(8) • • •• · · · •

• • •

(9) •...

• •• • · · · •

• •...

•...•

•

(10) •.

. .

••

• • • · · · ••

•.

..

•


(11) •...•• •• • · · · •• •...

•

(12) •...

•• • •

•• .

..

....

••

(13) •.

..

•

• • • •

•.

. .

•

(14) •

... •

• · · · •• • • •

•.

. .

•

(15) •.

..• •

• • ••.

. .

•

(16) •.

..•

•• • •

•.

. .

•

where any dashed line indicates a relation being the sum of all paths from the starting point to theend point. We note that the pg-critical algebras introduced above are strongly simply connected.


A crucial role in the proof of our main result will be played by the following characterizationof strongly simply connected algebras of polynomial growth established in [70, Theorem 4.1 andCorollary 4.2].

Proposition 1.4. Let A be a strongly simply connected algebra. The following conditions areequivalent:

(i) A is of polynomial growth.(ii) A does not contain a convex subcategory which is pg-critical or hypercritical.

(iii) The Tits form qA of A is weakly nonnegative and A does not contain a convex subcategorywhich is pg-critical.

A strongly simply connected algebra A is said to be extremal (see [9]) if there is an indecom-posable (finite dimensional) A-module M whose support suppM contains all extreme vertices(sinks and sources) of the Gabriel quiver QA of A. Observe that the convex hull of the supportof an indecomposable module over a strongly simply connected algebra is an extremal stronglysimply connected algebra. The following fact proved in [58, Theorem] (extending [55, Theorem]and [56, Theorem 1]) will be also essential for our considerations.

Theorem 1.5. Let A be a strongly simply connected algebra satisfying the following conditions:

(i) A is extremal;(ii) qA is weakly nonnegative;

(iii) A contains a convex subcategory which is either representation-infinite tilted algebra oftype Ep , p = 6,7,8, or a tubular algebra.

Then A is of polynomial growth.

As a direct consequence of Proposition 1.4 and Theorem 1.5 we obtain

Corollary 1.6. Let A be an extremal strongly simply connected algebra with weakly nonnega-tive Tits form qA and containing a pg-critical convex subcategory. Then every critical convexsubcategory of A is of type Dm, for some m � 4.

The following example shows that the Main Theorem of the paper cannot be extended toarbitrary simply connected algebras.

Example 1.7. Let A be the bound quiver algebra given by the quiver

7 2β

3γ

4σ

6ω

8 1η

ξ

5α

δ

bound by the relations αξ = 0, αη = δσγβη, ωσγβ = 0. Denote by B (respectively, H ) the fullsubcategory of A formed by the vertices 1, 2, 3, 4, 5 and 6 (respectively, 1, 2, 3, 4 and 5). Then B


is a one-point extension of the hereditary algebra H of Euclidean type A4 by an indecomposableregular module of regular length 3 lying in the unique stable tube of rank 4 of the Auslander–Reiten quiver ΓH , and hence B is wild, by [62, Theorem 3]. Therefore, A is also wild. Further,A is a triangular algebra with the separation property, and hence is simply connected. Clearly,A is not strongly simply connected, because the full convex subcategory H of A is not simplyconnected. On the other hand, the Tits form qA of A coincides with the Tits form qΛ of the boundquiver algebra Λ given by the quiver

7 2β

3γ

8 1η

ξ

5α

4δ

σ

6ω

bound by the relations αξ = 0, αη = 0, ωδα = ωσγβ . Denote by R the hereditary full subcate-gory of Euclidean type A4 of Λ formed by the vertices 1, 2, 3, 4 and 5. Then Λ can be obtainedfrom H by two one-point coextensions of R (with the coextension vertices 7 and 8) by the samesimple regular R-module lying in the unique stable tube of rank 2 of ΓR , and the one-point ex-tension (with extension vertex 6) by the simple regular R-module lying in a stable tube of rank 1of ΓR . Invoking again [62, Theorem 3], we conclude that Λ is tame (even one-parametric). Inparticular, we obtain that qA = qΛ is weakly nonnegative. Finally, we also note that Λ is simplyconnected but clearly not strongly simply connected.

2. Degenerations of algebras

For a positive integer d , we denote by algd(K) the affine variety of associative algebra struc-tures with identity on the affine space Kd . Then the general linear group GLd(K) acts on algd(K)

by transport of the structure, and the GLd(K)-orbits in algd(K) correspond to the isomorphismclasses of d-dimensional algebras (we refer to [38] for more details). We identify a d-dimensionalalgebra A with the point of algd(K) corresponding to it. For two d-dimensional algebras A

and B , we say that B is a degeneration of A (A is a deformation of B) if B belongs to the closureof the GLd(K)-orbit of A in the Zariski topology of algd(K).

C. Geiss’ Theorem [33] says that if A and B are two d-dimensional algebras, A degeneratesto B and B is a tame algebra, then A is also a tame algebra. We will apply this theorem in thefollowing special situation.

Proposition 2.1. Let d be a positive integer, and A(λ),λ ∈ K , be an algebraic family in algd(K)

such that A(λ) ∼= A(1) for all λ ∈ K \ {0}. Then A(1) degenerates to A(0). In particular, if A(0)

is tame, then A(1) is also tame.

A family of algebras A(λ), λ ∈ K , in algd(K) is said to be algebraic if the induced mapsA(−) :K → algd(K) is a regular map of affine varieties.

Following [73] an algebra A is said to be special biserial if A is isomorphic to a bound quiveralgebra KQ/I , where the bound quiver satisfies the conditions:

(a) each vertex of Q is a source and sink of at most two arrows,(b) for any arrow α of Q there are at most one arrow β and at most one arrow γ with αβ /∈ I

and γ α /∈ I .


The following fact has been proved in [75] (see also [20,24]).

Proposition 2.2. Every special biserial algebra is tame.

The aim of this section is to collect some results on degenerations of bound quiver algebraswhich will allow to show in Section 4 that certain glueings of critical algebras of types Dn

degenerate to special biserial algebras, and consequently are tame. In particular, we show here(Proposition 2.9) that every pg-critical algebra degenerates to a special biserial algebra.

We start with the following lemma proved in [19, Lemma 5.3].

Lemma 2.3. Let A = KQ/I be a bound quiver algebra whose quiver Q contains a convexsubquiver Q′ of the form

x1α1

y x2α2

where x1, x2 are sources of Q and α1 and α2 are unique arrows starting at x1 and x2, respec-tively. Assume that the ideal I admits a set R of generators of the form

R = {α1b1, . . . , α1bn,α2b1, . . . , α2bn, c1, . . . , cm}

with certain elements b1, . . . , bn ∈ ey(KQ) and c1, . . . , cm ∈ ez(KQ) for x1 �= z �= x2.Let A = KQ/I be the bound quiver algebra obtained from A as follows: the quiver Q is

obtained from Q by replacing the subquiver Q′ by the subquiver Q′ of the form

xεα

y

and I is the ideal of KQ generated by the set

R = {ε2, αb1, . . . , αbn, c1, . . . , cm

}.

Then A degenerates to A.

Lemma 2.4. Let A = KQ/I be a bound quiver algebra whose quiver Q contains a convexsubquiver Q′ of the form

y1β1

x

α1

α2

z

y2β2

and α1, α2, β1, β2 are unique arrows of Q having y1 and y2 as the ending or starting vertices,respectively. Assume that the ideal I admits a set R of generators of the form

R ={

c1α1, . . . , cnα1, c1α2, . . . , cnα2, β1d1, . . . , β1dm,

β d , . . . , β d ,α β + μα β ,w , . . . ,w

}
2 1 2 m 1 1 2 2 1 r


for some μ ∈ K \ {0}, certain elements c1, . . . , cn ∈ (KQ)ex , d1, . . . , dm ∈ ez(KQ) and w1 ∈eu1(KQ)ev1 , . . . ,wr ∈ eur (KQ)evr with u1, v1, . . . , ur , vr different from y1, y2. Let A = KQ/I

be the bound quiver algebra obtained from A as follows: the quiver Q is obtained from Q byreplacing the subquiver Q′ by the subquiver Q′ of the form

xα

y

ε

βz

and I is the ideal in KQ generated by the set

R ={

c1α, . . . , cnα,βd1, . . . , βdm,

ε2, αβ, w1, . . . , wr

}where w1, . . . , wr are obtained from w1, . . . ,wr by replacing any subpaths of the form α2β2(respectively, α1β1) by αεβ (respectively, −μαεβ). Then A degenerates to A.

Proof. For each λ ∈ K , consider the bound quiver algebra A(λ) = KQ/I (λ), where I (λ) is theideal in KQ generated by the set

R(λ) ={

c1α, . . . , cnα,βd1, . . . , βdm,

ε2 − λε,αβ, w1, . . . , wr

}.

Note that in case λ �= 0 the ideal I (λ) is not admissible since then the generator ε2 − λε is notcontained in the ideal of KQ generated by all paths of length two. Moreover, the quiver Q is notthe Gabriel quiver QA(λ) of A(λ) in this case. But A(λ), λ ∈ K , is a family of algebras of thesame dimension, depending algebraically on λ ∈ K . Clearly, A(0) = A. In order to prove thatA degenerates to A, it is enough to show that A ∼= A(λ) for λ ∈ K \ {0}. In order to simplifynotation, we will identify the elements of KQ (respectively, KQ) with their residue classes inKQ/I (respectively, KQ/I (λ)). We first show that A ∼= A(1). Consider the isomorphism ofalgebras f :A → A(1) which is defined by

f (ey1) = ε, f (α1) = αε, f (β1) = εβ,

f (ey2) = ey − ε, f (α2) = μ−1α(ey − ε), f (β2) = (ey − ε)β,

and f (ev) = ev , f (γ ) = γ , for the remaining idempotents ev and arrows γ of KQ. Observe that

αεβ = f (α1)f (β1) = f (α1β1) = −μf (α2β2)

= −μf (α2)f (β2) = −α(ey − ε)β,

and this is equivalent to αβ = 0, because ε2 = ε in A(1). Further, for i ∈ {1, . . . , n},0 = f (ciα1) = ciαε and 0 = f (ciα2) = μ−1ciα(ey − ε)

are equivalent to ciα = 0. Similarly, for j ∈ {1, . . . ,m},0 = f (β1dj ) = εβdj and 0 = f (β2dj ) = (ey − ε)βdj


are equivalent to βdj = 0. Finally, assume that ws , for some s ∈ {1, . . . , r}, is of the form pα1β1q

or pα2β2q . Hence pα1β1q = 0 = pα2β2q in A, because α1β1 = −μα2β2. But then

0 = f (pα1β1q) = p(αε)εβq

and

0 = f (pα2β2q) = μ−1pα(ey − ε)(ey − ε)βq = μ−1pα(ey − ε)βq

are equivalent to pαεβq = 0, because αβ = 0 and ε2 = ε in A(1). Therefore, f is a well-definedisomorphism of bound quiver algebras.

Fix now λ ∈ K \ {0} and consider the automorphism ψ :KQ → KQ defined by ψ(ε) =λ−1ε, and keeping the other paths (including those of length 0) unchanged. Observe that λ−1ε =ψ(ε) = ψ(ε2) = (λ−1ε)2 is equivalent to ε2 = λε, and ψ preserves the relations given by R(1) \{ε2 − ε} = R(λ) \ {ε2 − λε}. Therefore, ψ induces an algebra isomorphism A(1)

∼−→ A(λ). �Lemma 2.5. Let A = KQ/I be a bound quiver algebra whose quiver Q contains a convexsubquiver Q′ of the form

y1α2

y2 · · · ym−2αm−1

ym−1

αm

x

α1

β1

z

t

β2

with m � 2, and possibly some of the arrows α2, . . . , αm−1 are loops. Assume that the ideal I

admits a set R of generators of the form

R = {α1α2 · · ·αm + μβ1β2, c1, . . . , cn},

for some μ ∈ K \ {0}, satisfying one of the conditions:

(1) There is i ∈ {1, . . . ,m} such that αi occurs only in zero-relations of {c1, . . . , cn}.(2) There is j ∈ {1,2} such that βj occurs only in zero-relations of {c1, . . . , cn}.

Let A = KQ/I be the bound quiver algebra, where the ideal I is generated by the set

R = {β1β2, c1, . . . , cn},

where, for i ∈ {1, . . . , n}, ci = uα1α2 · · ·αn if ci = uβ1, ci = α1α2 · · ·αnv if ci = β2v, and ci = ci

in the remaining cases.Then A degenerates to A.


Proof. For λ ∈ K , consider the bound quiver algebra A(λ) = KQ/I (λ), where the ideal I (λ) isgenerated by the set

R(λ) = {α1α2 · · ·αm + λμβ1β2, c1, . . . , cn}.Clearly, A(λ), λ ∈ K , is an algebraic family of algebras of the same dimension, with A(1) = A

and A(0) = A. In order to prove that A degenerates to A, it is sufficient to show that A ∼= A(λ)

for λ ∈ K \ {0}. Fix λ ∈ K , and define the automorphism of algebras ϕ :KQ → KQ as follows:ϕ(αi) = λ−1αi and ϕ preserves the other arrows of Q if (1) holds, or ϕ(βj ) = λβj and ϕ pre-serves other arrows if (2) holds. Observe that ϕ preserves the generators c1, . . . , cn ∈ R, andhence induces the required isomorphism A → A(λ). �Lemma 2.6. Let A = KQ/I be a bound quiver algebra whose quiver Q contains a convexsubquiver Q′ of the form

xα

yβ

z.

Assume that the ideal I admits a set R of generators

R =

⎧⎪⎨⎪⎩a1αβb1, . . . , amαβbm

c1α, . . . , cpα,βd1, . . . , βdq

f1, . . . , fn

⎫⎪⎬⎪⎭such that αβ /∈ I , α or β occurs only in zero-relations of R, and f1, . . . , fn are not of the formaαβb, cα, or βd . Let A = KQ/I be the bound quiver algebra obtained from A as follows: thequiver Q is obtained from Q by replacing the subquiver Q′ by the subquiver Q′ of the form

xα

δ

yβ

z

and I is the ideal of KQ generated by the set

R =

⎧⎪⎪⎪⎨⎪⎪⎪⎩αβ,a1δb1, . . . , amδbm

c1α, . . . , cpα, c1δ, . . . , cpδ

βd1, . . . , βdq, δd1, . . . , δdq

f1, . . . , fn

⎫⎪⎪⎪⎬⎪⎪⎪⎭ .

Then A degenerates to A.

Proof. For λ ∈ K , consider the bound quiver algebra A(λ) = KQ/I (λ), where the ideal I (λ)

is generated by the set R(λ) obtained from R by replacing αβ by αβ − λδ, and keeping theremaining elements of R. Then A(λ), λ ∈ K , is an algebraic family of algebras of the samedimension and with A(0) = A. Observe also that A ∼= A(1), because δ = αβ in A(1). We showthat A(1) ∼= A(λ) for λ ∈ K \ {0}. It follows from our assumption on A that either α or β occursonly in zero-relations. We may assume (without loss of generality) that α has this property. Then,


for a fixed λ ∈ K \ {0}, a required algebra isomorphism A(1)∼−→ A(λ) is given by mapping α

into λ−1α and keeping the remaining arrows of Q unchanged. Therefore, we have A ∼= A(λ) forλ ∈ K \ {0} and A = A(0), and hence A degenerates to A. �Lemma 2.7. Let A = KQ/I be a bound quiver algebra whose quiver Q contains a convexsubquiver Q′ of the form

z

α2

vσ

x

α1

β1

γ1

tβ2

yδ

w

u1γ2

u2 ...γm−1

um−1

γm

with m � 2, and possibly some of the arrows γ2, . . . , γm−1 are loops. Assume that I admits a setof generators R of the form

R =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

α1α2 + β1β2 + μγ1 · · ·γm,σγ1, γmδ

a1σα1α2δb1 − c1, . . . , arσα1α2δbr − cr

d1α1, . . . , dsα1, d1β1, . . . , dsβ1

α2f1, . . . , α2fp,β2f1, . . . , β2fp

g1, . . . , gn

⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭where μ ∈ K \ {0}, aiσα1α2δbi and ci , 1 � i � r , are pairs of parallel paths, d1, . . . , ds ∈(KQ)ex , f1, . . . , fp ∈ ey(KQ), and g1 ∈ ev1(KQ)ew1 , . . . , gn ∈ evn(KQ)ewn with v1,w1, . . . ,

vn,wn different from x, y, z, t , u1, . . . , um−1. Let A = KQ/I be the bound quiver algebrawhere the ideal I is generated by the set R obtained from R by replacing α1α2 + β1β2 +μγ1 · · ·γm by α1α2 + β1β2, keeping the remaining elements of R, and adding the elementsd1γ1 · · ·γm, . . . , dsγ1 · · ·γm,γ1 · · ·γmf1, . . . , γ1 · · ·γmfp . Then A degenerates to A.

Proof. For λ ∈ K , let A(λ) = KQ/I (λ), where the ideal I (λ) is generated by the set R(λ)

obtained from R by replacing α1α2 + β1β2 by α1α2 + β1β2 + λμγ1 · · ·γm and keepingthe remaining elements of R. Then A(λ), λ ∈ K , is an algebraic family of algebras ofthe same dimension and with A(0) = A. In order to prove that A degenerates to A, it isenough to show that A ∼= A(λ) for λ ∈ K \ {0}. For a fixed λ ∈ K \ {0}, a required al-gebra isomorphism A → A(λ) is obtained by mapping σ into λσ , α1 into λ−1α1, β1 intoλ−1β1 and keeping the remaining arrows of Q unchanged. Note that, in A = KQ/I , therelations α1α2 + β1β2 + μγ1 · · ·γm = 0, d1α1 = 0, . . . , dsα1 = 0, d1β1 = 0, . . . , dsβ1 = 0,α2f1 = 0, . . . , α2fp = 0, β2f1 = 0, . . . , β2fp = 0, force the zero-relations d1γ1 · · ·γm = 0, . . . ,

dsγ1 · · ·γm = 0, γ1 · · ·γmf1 = 0, . . . , γ1 · · ·γmfp = 0. �Lemma 2.8. Let A = KQ/I be a bound quiver algebra whose quiver Q contains a convexsubquiver Q′ of the form


z

α2

x

α1

β1

γ1

tβ2

y

u1γ2

u2 ...γm−1

um−1

γm

with m � 2, and possibly some of the arrows γ2, . . . , γm−1 are loops. Assume that I admits a setof generators R of the form

R =

⎧⎪⎪⎨⎪⎪⎩α1α2 + β1β2 + μγ1 · · ·γm

d1α1, . . . , dsα1, d1β1, . . . , dsβ1

α2f1, . . . , α2fp,β2f1, . . . , β2fp

g1, . . . , gn

⎫⎪⎪⎬⎪⎪⎭where μ ∈ K \{0}, d1, . . . , ds ∈ (KQ)ex , f1, . . . , fp ∈ ey(KQ), and g1 ∈ ev1(KQ)ew1 , . . . , gn ∈evn(KQ)ewn with v1,w1, . . . , vn,wn different from x, y, z, t , u1, . . . , um−1. Let A = KQ/I bethe bound quiver algebra where the ideal I is generated by the set R obtained from R by re-placing α1α2 + β1β2 + μγ1γ2 · · ·γm by α1α2 + β1β2, keeping the remaining elements of R, andadding the elements d1γ1 · · ·γm, . . . , dsγ1 · · ·γm,γ1 · · ·γmf1, . . . , γ1 · · ·γmfp . Then A degener-ates to A.

Proof. Similar to the proof of Lemma 2.7. �An essential role in the proof of our main result will be played by the following proposition.

Proposition 2.9. Every pg-critical algebra degenerates to a special biserial algebra.

Proof. This follows by suitable iterated applications of Lemmas 2.3–2.8 (and their duals) to anyof the 16 families of pg-critical algebras. We present all necessary degeneration procedures forsome pg-critical algebras of types (3), (10) and (16).

Let A be the pg-critical algebra of type (3) of the form

•• .

.

.• · · · • •

• • • · · · •• •...

•Applying the dual of Lemma 2.3 and Lemmas 2.5 and 2.6, we degenerate A to the special biserialalgebra given by the quiver


•ξ

• α ...•

β

· · · •γ• •

σ•

η

· · · • • ε

•...

•bound by the relations αβ = 0, γ σ = 0, ξη = 0, ε2 = 0.

Let A be the pg-critical algebra of type (10) of the form

• •

. .. •

•• .

..

.• • • · · · ••

... •

•• •

Then, applying Lemmas 2.4 and 2.5, we degenerate A to the special biserial algebra given by thequiver

•

γ

•

ξ..

. ••

•α

.

..

.•ε

β

•

σ

· · · •

η

•

... •

•• •

bound by the relations ε2 = 0, αβ = 0, γ σ = 0 and ξη = 0.


Finally, let A be the pg-critical algebra of type (16). Applying Lemma 2.8, we degeneratefirst A to the algebra B given by the quiver

•

σ1 σ2

.

..

•γ

•β

•

�1

•

�2

•α

•.

..

•

bound by σ1�1 + σ2�2 = 0, γβα = 0. Then, applying Lemmas 2.4 and 2.6, we degenerate B tothe special biserial algebra C given by the quiver

•

σ

.

..

•γ

δ•β

•ε

�

•α

•.

..

•

bound by the relations ε2 = 0, σ� = 0, γβ = 0, δα = 0. �3. Coil algebras

The aim of this section is to recall the concept of coil algebras, which plays a fundamentalrole in the proof of our main theorem.

We use freely properties of the Auslander–Reiten quiver ΓA of an algebra A, for which werefer to [6] and [63]. We agree to identify the vertices of ΓA with the corresponding indecompos-able A-modules. A component Γ of ΓA is said to be standard if the full subcategory of modA

formed by the indecomposable modules from Γ is equivalent to the mesh category K(Γ ) of Γ

(see [13,63]). Recall also that a stable tube (of rank r � 1) of ΓA is a component of the formZA∞/(τ r ).


Given a standard component Γ of ΓA and an indecomposable module X in Γ , the supportS(X) of the functor HomA(X,−)|Γ is the K-linear category defined as follows (see [4]). LetHX denote the full subcategory of modA formed by the indecomposable modules M in Γ suchthat HomA(X,M) �= 0, and JX denote the ideal of HX consisting of the morphisms f :M → N

(with M , N in HX) such that HomA(X,f ) = 0. We define S(X) to be the quotient categoryHX/JX .

Let A be an algebra and Γ be a standard component of ΓA. For an indecomposable mod-ule X in Γ , called the pivot, three admissible operations are defined, depending on the supportS(X) of the functor HomA(X,−)|Γ . These admissible operations yield in each case a modifiedalgebra A′, and a modified component Γ ′ of Γ (see [3] for more details):

(ad 1) If S(X) is the path category of the infinite linear quiver

X = X0 −→ X1 −→ X2 −→ · · ·

X is called an (ad 1)-pivot, and we set A′ = (A × D)[X ⊕ Y1], where D is the full t × t uppertriangular matrix algebra (with t � 1), and Y1 is the unique indecomposable projective–injectiveD-module. In this case, Γ ′ is obtained by inserting in Γ a rectangle consisting of the modulesZij = (K,Xi ⊕Yj ,

( 11

)) for i � 0, 1 � j � t , and X′

i = (K,Xi,1) for i � 0, where Yj , 1 � j � t ,denote the indecomposable injective D-modules. If t = 0, we set A′ = A[X] and the rectanglereduces to the ray formed by modules of the form X′

i .(ad 2) If S(X) is of the form

Yt ←− · · · ←− Y1 ←− X = X0 −→ X1 −→ X2 −→ · · ·

with t � 1 (so that X is injective), X is called an (ad 2)-pivot, and we set A′ = A[X]. In this case,Γ ′ is obtained by inserting in Γ a rectangle consisting of the modules Zij = (K,Xi ⊕ Yj ,

( 11

))

for i � 0, 1 � j � t , and X′i = (K,Xi,1) for i � 0.

(ad 3) If S(X) is the bound quiver category of a quiver of the form

Y1 Y2 · · · Yt−1 Yt

X = X0 X1 · · · Xt−2 Xt−1 Xt Xt+1 · · ·

with t � 2 (so that Xt−1 is injective), bound by the mesh relations of the squares, X is calledan (ad 3)-pivot, and we set A′ = A[X]. In this case, Γ ′ is obtained by inserting in Γ a rectangleconsisting of the modules Zij = (K,Xi ⊕ Yj ,

( 11

)) for i � 1, 1 � j � i, and X′

i = (K,Xi,1) fori � 0.

It was shown in [3] that Γ ′ is a standard component of ΓA′ containing the module X. The dualcoextension operations (ad 1∗), (ad 2∗), (ad 3∗) are also called admissible. A translation quiver Cis called a coil if there exists a sequence of translation quivers Γ0,Γ1, . . . ,Γn = C such that Γ0 isa stable tube and, for each 0 � i < n, Γi+1 is obtained from Γi by an admissible operation [3].

Let C be a critical algebra (preprojective tilt of a hereditary algebra of type Dn, E6, E7, or E8)and T be the P1(K)-family of standard stable tubes in ΓC . Following [5] an algebra B is calleda coil enlargement of C if there is a finite sequence of algebras C = A0,A1, . . . ,Am = B suchthat, for each 0 � j < m, Aj+1 is obtained from Aj by an admissible operation with pivot orcopivot in a stable tube of T or in a coil ΓA , obtained from a stable tube of T by means of the

j


sequence of admissible operations done so far. A distinguished property of a coil enlargement ofa critical algebra is the existence of a P1(K)-family of standard coils. We refer to [41, Section 3]for a description of indecomposable modules lying in standard coils.

Recall also that a tubular extension (respectively, tubular coextension) of C in the sense ofC.M. Ringel [63, (4.7)] is a coil enlargement B of C such that each admissible operation in thesequence defining it is of type (ad 1) (respectively, (ad 1∗)). An essential role in our considera-tions will be played by the following structure result proved in [5, Theorem 3.5].

Proposition 3.1. Let B be a coil enlargement of a critical algebra C. Then:

(i) There is a unique maximal tubular coextension B− of C which is a convex subcategory of B ,and B is obtained from B− by a sequence of admissible operations of types (ad 1), (ad 2),(ad 3).

(ii) There is a unique maximal tubular extension B+ of C which is a convex subcategory of B ,and B is obtained from B+ by a sequence of admissible operations of types (ad 1∗), (ad 2∗),(ad 3∗).

(iii) Every object of B belongs to B− or B+.

We note that the bound quiver of a tubular extension (respectively, tubular coextension) B ofa critical algebra C is obtained from the bound quiver of C by adding a finite family of branchesat the extension vertices of one-point extensions (respectively, at the coextension vertices of one-point coextensions) of C by simple regular modules. Recall that a branch [63, (4.4)] is a finiteconnected bound subquiver of the following infinite bound quiver, containing the root b,

· · · · · · · · · · · ·

• • • • • • • •

• • • •

• •

•b

where the dashed lines denote the zero-relations of length 2. We also note that the class of boundquiver algebras of branches coincides with the class of tilted algebras of the hereditary algebrasgiven by the equioriented quivers • → • → · · · → • → • of types Am, m � 1 (see [63, Proposi-tion 4.4(2)]). Finally, we point out that the bound quiver algebra of a branch is a strongly simplyconnected representation-finite special biserial algebra, and hence the support of any of its inde-


composable modules is the path algebra of a linear quiver (usually with many sources and sinks)of type An, n � 1 (see [73]).

It follows from [5, Corollary 4.2] that a coil enlargement B of a critical algebra C is tame ifand only if the Tits form qB is weakly nonnegative. A tame coil enlargement of a critical algebrais called a coil algebra. The Auslander–Reiten quiver ΓA of a coil algebra A consists of a pre-projective component, a preinjective component and infinitely many coils (see [5, Theorem 3.5,Corollary 4.2]). Moreover, all coil algebras are strongly simply connected algebras of polynomialgrowth. The following proposition shows the importance of coil algebras in the representationtheory of strongly simply connected algebras of polynomial growth (see [70, Corollary 4.8]).

Proposition 3.2. Let A be a strongly simply connected algebra of polynomial growth. Then thereexist convex coil subcategories B1, . . . ,Bm of A whose indecomposable modules exhaust all butfinitely many isoclasses of indecomposable A-modules. Moreover, the supports of the remainingfinitely many indecomposable A-modules are tame tilted convex subcategories of A.

4. DDD-algebras

In the study of non-polynomial growth tame strongly simply connected algebras a fundamen-tal role is played by some enlargements of critical algebras of types Dn, n � 1 (see Corollary 1.6).Recall from [11] and [34] that there are only four families of critical algebras of types Dn, n � 1,given by the following bound quivers

• •

• • · · · • •

• •

• •.

..

.• • · · · • •

• •

• •.

..

.

.

..

.• • · · · • •

• •

• · · · •• •

••

where the number of vertices is equal n + 1 and • — • means • → • or • ← •. It is well known(see [63, (4.3)]) that the Auslander–Reiten quiver ΓC of a critical algebra C of type Dn consistsof a preprojective component, a preinjective component, and a P1(K)-family of standard stabletubes, two of them of rank 2, one of rank n − 2, and the remaining ones of rank 1. Observe that,for n = 4, ΓC has 3 stable tubes of rank n − 2 = 2.

In the paper, by a D-coil algebra is meant a coil enlargement B of a critical algebra C oftype Dn using modules from a fixed stable tube of rank n − 2 in ΓC . It follows from [5, Theo-rem 4.1, Corollary 4.2] and results of [63, (4.9)] that the Auslander–Reiten quiver ΓB of a D-coil


algebra B consists of a preprojective component, a preinjective component, a K-family of stan-dard stable tubes (two tubes of rank 2, the remaining ones of rank 1), and a standard coil havingat least n − 2 rays and at least n − 2 corays, and usually many projective modules and manyinjective modules. This coil will be called the large coil of ΓB . Observe that the large coil of ΓB

is uniquely defined, except the case B = C is a critical algebra of type D4. In this case, by thelarge coil we mean a fixed stable tube of rank 2 of ΓC . Clearly, a D-coil algebra is a coil algebra.We also note that every D-coil algebra contains exactly one critical convex subcategory, and is aglueing of two representation-infinite tilted algebras of (usually different) types Dr , r � 4.

In order to define the class of D-algebras we need also the concepts of D-extensions and D-coextensions of D-coil algebras. Suppose A and A′ are two algebras (considered as K-categories)containing a common convex subcategory B . Then we denote by Λ = A�

BA′ the pushout A

and A′ along the embeddings of B into A and A′. Observe that the quiver QΛ of Λ is obtainedby glueing the quivers QA and QA′ along the quiver QB , and the ideal defining Λ is the ideal inthe path algebra KQΛ generated by the ideals defining the algebras A and A′.

Let B be a D-coil algebra and Γ a large coil of ΓB . By a D-extension of B we mean a stronglysimply connected algebra of one of the forms:

(d1) B[X] �Kω

H , where X is an indecomposable module in Γ such that the support S(X) of

the functor HomB(X,−)|Γ is the path category of the linear quiver

X = X0 −→ X1 −→ X2 −→ · · ·

H is the path algebra of a quiver �(m) of the form

b

ω = a1 a2 · · · am

c

m � 1, and Kω = K is the simple algebra given by the extension vertex ω of B[X] and the uniquesource ω = a1 of �(m);

(d2) B[X], where X is an indecomposable module in Γ and the support S(X) ofHomB(X,−)|Γ is the bound quiver category of the quiver

Y1 Y2 Y3 · · ·

X = X0 X1 · · · Xt Xt+1 Xt+2 · · ·

with t � 0, bound by the mesh relations of the squares;(d3) B[X], where X is an indecomposable module in Γ and the support S(X) of

HomB(X,−)|Γ is the bound quiver category of the quiver


Y2

Y1 Z1

X = X0 X1 X2 X3 · · ·

bound by the mesh relation of the unique square;(d4) B[X], where X is an indecomposable module in Γ and the support S(X) of

HomB(X,−)|Γ is the bound quiver category of the quiver

Y2 Z2

Y1 Z1

X = X0 X1 X2 X3 · · ·

bound by the mesh relations of the two squares.A D-coextension of B is defined dually invoking the dual coextension constructions (d1∗),

(d2∗), (d3∗), (d4∗). Since the class of D-coil algebras is closed under making the opposite al-gebras, we conclude that the class of D-coextensions of D-coil algebras coincides with theclass of opposite algebras of D-extensions of D-coil algebras. We would like to mention thatD-extensions of types (d1) and (d2) were applied in [46] to define the pg-critical algebras. Infact, it is rather easy to see that every D-extension (respectively, D-coextension) A of a D-coilalgebra B creates a new critical algebra of type Dn, which can be used to create new D-coilalgebras and their D-extensions or D-coextensions. Finally, we mention that in general the one-point extensions of type (d2) (respectively, the one-point coextensions of type (d2∗)) may containconvex hereditary subcategories of type Am, and hence they are not strongly simply connected(see the algebras of types (17)–(31) in [46, Theorem 3.2]). Therefore, the assumption that aD-extension (respectively, D-coextension) is strongly simply connected is essential for our con-siderations.

We need also the concept of a blowup of an algebra. Let A = KQ/I be a bound quiveralgebra. A vertex a of Q is said to be narrow if the quiver Q of A contains a convex subquiver �

of the form

xα

aβ

y

with αβ /∈ I , and α (respectively, β) is the unique arrow of Q ending (respectively, starting) at a.For a narrow vertex a of Q, we define the blowup A〈a〉 = KQ〈a〉/I 〈a〉 of A at the vertex a asfollows. The quiver Q〈a〉 is obtained from the quiver Q by replacing the subquiver � by thesubquiver �〈a〉 of the form

a1β1

x

α1

α2

y

a2β2


and keeping the remaining vertices and arrows of Q unchanged. Then the ideal I 〈a〉 of KQ〈a〉 isobtained from the ideal I of KQ by adding the generator α1β1 −α2β2, replacing any generator ofthe form uα by two generators uα1 and uα2, any generator of the form βv by two generators β1v

and β2v, any generator containing αβ by the generator with αβ replaced by α1β1, and keepingthe remaining generators of I unchanged. Further, a set S of narrow vertices of Q is said to beorthogonal if Q does not admit an arrow connecting two vertices of S. By a blowup of A wemean an iterated blowup A〈a1, . . . , ar〉 = A〈a1〉〈a2〉 · · · 〈ar 〉 of A with respect to an orthogonalset a1, . . . , ar of narrow vertices of Q.

We are now in position to give a recursive definition of a D-algebra:

(i) All D-coil algebras are D-algebras;(ii) All D-extensions and D-coextensions of D-coil algebras are D-algebras;

(iii) Suppose A is a D-algebra and contains a D-coil algebra B as a convex subcategory. Let A′be a D-extension or a D-coextension of B , or a D-coil algebra containing B as a convexsubcategory. Then the pushout Λ = A�

BA′ is a D-algebra provided it does not contain a hy-

percritical convex subcategory (equivalently, the Tits form qΛ of Λ is weakly nonnegative);(iv) All blowups of D-algebras are D-algebras.

We would like to mention that there is a complete local understanding of the bound quiverpresentations of D-algebras. Namely, by Proposition 3.1, every D-coil algebra B is a suitableglueing of a tubular extension B+ and a tubular coextension B− of the same critical algebra C

of type Dn. Moreover, by [63, (4.7)], the tubular extensions (respectively, coextensions) of thecritical algebras C are obtained from C by adding branches (in the sense of [63, (4.4)]) at theextension (respectively, coextension) vertices of the one-point extensions (respectively, coex-tensions) of C by the applied simple regular C-modules. Further, a complete description of allsimple regular modules and all indecomposable regular modules of regular length 2 (appliedin the D-extensions and D-coextensions) over the critical algebras of types Dn is given in [45,Section 2]. Finally, the forbidden hypercritical algebras are described by quivers and relationsin [39,74,76].

We exhibit the following properties of D-algebras which will be essential in further consider-ations.

Proposition 4.1. Let A be a D-algebra. Then:

(i) A is strongly simply connected.(ii) Aop is a D-algebra.

(iii) Every object a of A is an object of a convex subcategory Λ of A which is a tubular extensionor a tubular coextension of a critical convex subcategory C.

(iv) Every object a of A is an object of a convex D-coil subcategory B of A.

Proof. The properties (i) and (ii) are consequences of the definition of a D-algebra. Further, byProposition 3.1, (iv) implies (iii). We show now that (iv) also holds. Indeed, the new vertices ofany D-extension (respectively, D-coextension) of a D-coil algebra inside A belong to the creatednew critical category. Finally, for any blowup inside A, the new two objects (say a1 and a2),replacing an old narrow object a, belong to a new critical convex subcategory. We also note thatthe blowups of D-coil algebras B usually change the tubular extensions and tubular coextensionsto which the objects of B belong. �


The statement (iv) says that every D-algebra A admits a finite family of convex D-coil subcat-egories which together exhaust all objects of A (an atlas of A by D-coil algebras). We also notethat the class of algebras which are tubular extensions or tubular coextensions of critical algebrasand occur as convex subcategories of D-algebras coincides with the class of all strongly simplyconnected representation-infinite tilted algebras of types Dn, n � 4 (see also [63, (4.9)]). There-fore, the statement (iii) of Proposition 4.1 can be reformulated as follows: every D-algebra A

admits an atlas formed by convex subcategories which are representation-infinite tilted algebrasof type Dn and together exhaust all objects of A (an atlas of A by representation-infinite tiltedalgebras of types Dn).

The following example illustrates the above considerations.

Example 4.2. Let B be the algebra given by the bound quiver

13

11 12 5

1 10 6

19

3 4 7 18

2 8 17

14

9

15 16

We claim that B is a D-coil algebra. Denote by C the critical convex subcategory of B of type D8

given by the objects 1, 2, 3, 4, 5, 6, 7, 8, 9. Then B is a coil enlargement of C by four admissibleoperations of type (ad 1), creating the sets of vertices {10}, {11,12}, {13}, {18}, three admissibleoperations of type (ad 1∗), creating the sets of vertices {14}, {15}, {16,17}, and one admissibleoperation of type (ad 2), creating the vertex 19. Moreover, B is a D-coil algebra because only thesimple regular C-modules (the simple modules SC(3) and SC(8) at the vertices 3 and 8) fromthe unique (large) stable tube of rank 6 of ΓC are used. In the notation of Proposition 3.1, themaximal tubular coextension B− of C is the convex subcategory of B given by the objects of C

and the objects 14, 15, 16 and 17, while the maximal tubular extension B+ of C is the convexsubcategory of B given by the objects of C and the objects 10, 11, 12, 13, 17, 18 and 19. We alsonote that the object 17 belongs to B− and B+. Consider now the algebra A given by the boundquiver


13

11 12 5 20

10′ 10′′ 6

19

1 3 4 7 18

2 8 17′ 17′′

14

9

15 16

Then A is a D-algebra, obtained from the D-coil algebra B by D-extension, creating the ver-tex 20, and two blowups at the vertices 10 and 17, creating the sets of vertices {10′,10′′} and{17′,17′′}. Observe that A contains five pairwise different critical convex subcategories: the cat-egory C = C1, the category C2 given by the objects 1, 2, 3, 10′ and 10′′, the category C3 givenby the objects 1, 2, 3, 4, 5, 6, 7 and 20, the category C4 given by the objects 1, 2, 3, 4, 5, 6, 7,8, 17′, 17′′, 18 and 19, and the category C5 given by the objects 1, 2, 3, 4, 8, 9, 16, 17′ and 17′′.We note that in A the vertices 11, 12, 13 form the branch of a tubular extension of the criticalcategory C2, and do not belong to a tubular extension of the critical subcategory C. Further,the convex subcategory B1 of A given by the objects 1, 2, 3, 10′, 10′′, 11, 12, 13, 14 and 15is a D-coil algebra, which is the coil enlargement of the critical algebra C2 by two admissibleoperations of type (ad 1), creating the sets of vertices {11,12}, {13}, and two admissible oper-ations of type (ad 1∗), creating the sets of vertices {14}, {15}. Clearly, the objects 14, 15 andthe objects of C form another convex D-coil subcategory of A. Finally, observe that if we takethe blowup Λ = B〈6,10,17〉 of B at the pairwise orthogonal narrow vertices 6, 10, 17, thenΛ is a D-algebra which does not contain the unique critical subcategory C of B as a convexsubcategory.

The main aim of this section is to prove the following theorem.

Theorem 4.3. Let A be a D-algebra. Then A is a tame algebra.

The proof of the above theorem is divided into three main steps. The third step will consistof degenerations of certain D-algebras to special biserial algebras, with application of resultsdescribed in Section 2. In the first two steps we will remove obstructions which do not allowapply the degeneration results collected in Section 2 to arbitrary D-algebras.

We need a preliminary result on some special one-point extension algebras.


Lemma 4.4. Let B be an algebra, Γ a component of ΓB and X an indecomposable B-modulelying in Γ . Assume that the support S(X) of HomB(X,−)|Γ admits a convex subcategory givenby the quiver

Y2 Z2

Y1 Z1

X = X0 X1 X2

bound by the mesh relations of the squares, possibly with Z2 = 0, such that the remaining objectsof S(X) are successors of X2. Then the triples

X′0 = (K,X0,1), X′

1 = (K,X1,1), U11 =(

K,X1 ⊕ Y1,

(11

)), and

U12 =(

K,X1 ⊕ Y2,

(11

)),

are the unique indecomposable B[X]-modules (V ,M,ϕ) with ϕ nonzero and M having a directsummand isomorphic to X0 or X1.

Proof. Observe that X′0 is the indecomposable projective B[X]-module P(ω) given by the

extension vertex ω of B[X] and with radP(ω) = X0. Then, applying the general theory of one-point extension algebras (see [63, (2.5)], [65, (17.3)]), we deduce that the neighborhood of P(ω)

in the Auslander–Reiten quiver ΓB[X] is as follows

Y2 X′1

Y1 U12 T

X = X0 P(ω) U11 Z1 R .. . . .

X1 U21 . . .

X2 . . .

. . .

where


U21 =(

K,X2 ⊕ Y1,

(11

)), R =

(K,X2 ⊕ Y2 ⊕ Z1,

(111

)),

T =(

K,X2 ⊕ Z1,

(11

)),

and we identify a B-module N with the triple (0,N,0). Let (V ,M,ϕ) be an indecomposableB[X]-module with ϕ :V → HomB(X,M) nonzero and the B-module M having a decompositionM = M ′ ⊕ M ′′, M ′ isomorphic to X0 or X1. Since (V ,M,ϕ) is indecomposable, we knowthat the composition of ϕ with the canonical projection HomB(X,M) → HomB(X,M ′) is alsononzero. Further, the spaces HomB(X,X0) and HomB(X,X1) are one-dimensional. Hence thereis a commutative diagram of K-vector spaces

K1

f

HomB(X,X0)

HomB(X,g)

Vϕ

HomB(X,M)

where g :X0 → M is a map in modB with g(X0) = M ′. It follows from the shape of S(X)

that X1 is an indecomposable injective B-module, the simple socle SocX1 of X1 is isomorphicto a direct summand S of the socle SocX0 of X0, and for any nonzero map h :X0 → X1 inmodB its restriction to S defines an isomorphism S

∼−→ SocX1. In particular, the restrictionof g to S is nonzero. We also note that X′

1 is an injective B[X]-module. Suppose now thatthe indecomposable B[X]-module (V ,M,ϕ) is not isomorphic to one of the modules P(ω) =X′

0,U11,U12, or X′1. Then the nonzero map (f, g) : (K,X0,1) → (V ,M,ϕ) factors through a

direct sum of modules isomorphic to Z1, U21, R, T . But then g :X0 → M factors through adirect sum of the modules isomorphic to Z1, X2. This is a contradiction, because for any mape :X0 → Z1 ⊕ X2 we have e(S) = 0, and hence also g(S) = 0. This proves the lemma. �

A D-algebra A is said to be mild if, in the D-extensions and D-coextensions of D-coil algebrasapplied to obtain A, the procedures (d3), (d4), (d3∗) and (d4∗) are not involved. The followingproposition is our first reduction step.

Proposition 4.5. Let A be a D-algebra. Then there are a mild D-algebra A′, two full cofinitesubcategories X of indA′ and Y of indA, and a functor F : modA′ → modA such that:

(1) F is exact and preserves indecomposable modules;(2) F defines a functor X → Y which is dense and reflects isomorphisms.

Moreover, if A′ is tame then A is tame.

Proof. If A is a mild D-algebra, we take A = A′, X = Y = indA and F the identity functor.Assume A is not mild. In order to define the required mild D-algebra A′, we will replace eachof the operations of types (d3), (d4), (d3∗), (d4∗), involved in the definition of A, by a suitableoperation of one of the types (d1), (d2), (d1∗), (d2∗).

We divide the proof into several steps.


(1) Assume A contains a convex subcategory Ω which is a blowup of a one-point exten-sion B[X], where B is a D-coil algebra and X is an indecomposable B-module in the largecoil Γ of ΓB with S(X) of the form (d3):

Y2

Y1 Z1

X = X0 X1 X2 X3 · · ·

bound by the mesh relation of the square. Then Ω admits a convex subcategory Λ = D[X] ofthe form

ω

σ2

σ1

ξη

C

b

α β

c

γ

δ2δ1

a d

where possibly σ1 = σ2, or δ1 = δ2, ω is the extension vertex of the one-point extension D[X],D is a blowup of a D-coil convex subcategory of B , X = radPD(ω) = radPA(ω) is indecompos-able, ξβ = ηγ �= 0, ξα = σ1uδ1 �= 0 for a subpath u of QC , and αϕ = 0, γψ = 0, for possiblearrows ϕ, ψ in QA starting respectively from a and d . Moreover, α and β (respectively, β and γ )are unique arrows of QA starting at b (respectively, ending at d). Denote by Γ the componentof ΓD containing the module X. Then the support S(X) of the functor HomD(X,−)|Γ admits aconvex subcategory given by the quiver

Y2

Y1 Z1

X = X0 X1 X2

bound by the mesh relation of the square, and the remaining objects of S(X) are successorsof X2. Here, Y1 = ID(d) = IB(d) = IA(d), Y2 = ID(c) = IB(c) = SA(c), Z1 = ID(b) = IB(b) =


SA(b), and X1 = ID(a) = IA(a) with X1/Soc X1 ∼= Z1 ⊕ X2. Consider the modified category Λ′obtained from Λ by splitting at the objects b, c, d as follows

ωσ2

σ1 ξ

η

b1

β1

c1

γ1C

d1

δ2δ1

b2 = b

α2=α

β2=β

c2 = c

γ2=γ

a d2 = d

with ξβ1 = ηγ1 �= 0, and keeping the remaining parts of Λ unchanged.We study the relationship between the categories modΛ and modΛ′. Since Λ = D[X], we

may identify modΛ with the category of triples (V ,M,ϕ), where V is in modK , M is in modD,and ϕ :V → HomD(X,M) is a K-linear map. Observe that X0 and X1 are the unique indecom-posable D-modules L such that HomD(X,L) �= 0 and Lα �= 0. Applying Lemma 4.4, we thenconclude that the indecomposable Λ-modules

PΛ(ω) = X′0 = (K, X0,1), IΛ(a) = X′

1 = (K, X1,1),

U11 =(

K,X1 ⊕ Y1,

(11

)), U12 =

(K,X1 ⊕ Y2,

(11

)),

are the unique indecomposable Λ-modules N with Nξα �= 0.We identify modΛ and modΛ′ with the categories of finite dimensional representations of the

bound quivers defining respectively Λ and Λ′, and consider the canonical functor FΛ : modΛ′ →modΛ which assigns to any Λ′-module M ′ the Λ-module M as follows:

M(b) = M ′(b1) ⊕ M ′(b2), M(c) = M ′(c1) ⊕ M ′(c2),

M(d) = M ′(d1) ⊕ M ′(d2),

M(x) = M ′(x) for the remaining vertices x of QΛ′ ,

M(β) =(

M ′(β1) 00 M ′(β )

), M(γ ) =

(M ′(γ1) 0

0 M ′(γ )

),

2 2


M(ξ) =(

M ′(ξ)

0

), M(η) =

(M ′(η)

0

),

M(α) = (0,M ′(α)

),

M(�) = (0,M ′(�)

)for any arrow � starting at d = d2,

M(σ) = M ′(σ ) for the remaining arrows σ of QΛ.

Observe that FΛ is exact and preserves indecomposable modules. Denote by XΛ the fullsubcategory of indΛ′ given by all modules except the six modules having support in thefull subcategory of Λ′ given by b1, c1 and d1. Further, denote by YΛ the full subcate-gory of indΛ given by all modules except X′

0, X′1, U11 and U12, described above. Then the

restriction of FΛ to XΛ defines a functor XΛ → YΛ which is dense and reflects isomor-phisms. The above splitting Λ′ of Λ induces the corresponding splitting of the category Ω

to the category Ω ′ which is a blowup of a D-coil algebra B ′ defined as follows. It followsfrom the shape of S(X) that the large coil Γ of ΓB admits a translation subquiver of theform

Y2

τBY2 Y1 Z1

τ 2BY2 τBY1 X0 X1 X2 · · ·

Take U = τ 2BY2. Then the support S(U) of HomB(U,−)|Γ is of the form

U = U0 U1 U2 U3 · · ·

with U1 = τBY1 and Un = Xn−2 for n � 2, and hence U can be the pivot of an admissible op-eration of type (ad 1). Observe that U is an indecomposable B-module which occurs as the leftterm of a short exact sequence

0 −→ U −→ X −→ IB(d) −→ 0.

Consider now the coil enlargement B ′ = (B × E)[U ⊕ P ] of B of type (ad 1) with the pivot U ,E the algebra of 2 × 2 upper triangular matrices, and P the unique indecomposable projective–injective E-module. Then the blowup of B inside Ω induces the corresponding blowup Ω ′′ of B ′containing a convex subcategory Λ′′ of the form


ω

σ2

σ1ξ

b1

β1

Cd1

δ2δ1

b2 = b

α2=α

β2=β

c2 = c

γ2=γ

a d2 = d

Observe that Λ′ is the blowup of Λ′′ at the vertex b1. Then Ω ′ is the blowup of Ω ′′ at thevertex b1. In this modification process the blowup Ω of the D-extension B[X] of type (d3) isreplaced by the blowup Ω ′ of an admissible extension B ′ of B of type (ad 1).

Since Λ = D[X] is a convex subcategory of A, we may identify modΛ with the full subcat-egory of modA consisting of modules having support in Λ. We claim now that the indecom-posable Λ-modules X′

0, X′1, U11 and U12 are the unique indecomposable A-modules N with

Nξα �= 0. Indeed, by Proposition 1.2, there is a sequence Λ = Λ0,Λ1, . . . ,Λm = A of convexsubcategories of A such that for each i ∈ {0, . . . ,m − 1}, Λi+1 is a one-point extension or co-extension of Λi by an indecomposable Λi -module Ri . Hence, in order to prove our claim, it isenough to show that if Λi+1 = Λi[Ri] (respectively, Λi+1 = [Ri]Λi ) then HomΛi

(Ri,Z) = 0(respectively, HomΛi

(Z,Ri) = 0) for Z ∈ {X′0, X

′1,U11,U12}. Suppose Λi+1 = Λi[Ri] and

HomΛi(Ri,Z) �= 0 for Z ∈ {X′

0, X′1,U11,U12}. Then Ri admits a factor module which is iso-

morphic to a nonzero submodule of Z. But then a simple analysis of the known supports ofthe modules X′

0, X′1, U11 and U12 shows that Λi+1 (and hence A) contains a convex hyper-

critical subcategory of type ˜Dn, and this contradicts our assumption that A is a D-algebra.

Similarly, if Λi+1 = [Ri]Λi and HomΛi(Z,Ri) �= 0 for some Z ∈ {X′

0, X′1,U11,U12} then, since

Z/ radZ ∼= SA(ω), we infer that the vertex ω belongs to the support of Ri . But then we infer im-

mediately that Λi+1 (and hence A) admits a convex hypercritical subcategory of type ˜D5, again

a contradiction. This shows our claim.(2) Assume now that A contains a convex subcategory Ω which is a blowup of a one-point

extension B[X], where B is a D-coil algebra and X is an indecomposable B-module in the largecoil Γ of ΓB with S(X) of the form (d4):

Y2 Z2

Y1 Z1

X = X0 X1 X2 X3 · · ·


bound by the mesh relations of the squares. Then Ω admits a convex subcategory Λ = D[X] ofthe form

ω

σ2

σ1

ξ

C

δ2δ1

c

β

γ

b

α

d

a

where possibly σ1 = σ2, or δ1 = δ2, ω is the extension vertex of the one-point extension D[X],D is a blowup of a D-coil convex subcategory of B , X = radPD(ω) = radPA(ω), c

γ—–d

means cγ←−− d (and then γβα �= 0), or c

γ−−→ d (and then ξγ �= 0), ξβα = σ1uδ1 �= 0 for asubpath u of QC , and αϕ = 0, βψ = 0, for the possible arrows ϕ, ψ in QA starting respec-tively from a and b. Moreover, β , γ , ξ are the unique arrows connected to c, and γ is theunique arrow connected to d . Denote by Γ the component of ΓD containing the module X.Then the support S(X) of the functor HomD(X,−)|Γ admits a convex subcategory given by thequiver

Y2 Z2

Y1 Z1

X = X0 X1 X2

bound by the mesh relations of the squares, and the remaining objects of S(X) are successors ofX2. Here, X1 = ID(a) with X1/Soc X1 ∼= Z1 ⊕ X2, Z1 = ID(b) = IB(b), Z2 = ID(c) = IB(c),Y1 = PD(c)/SD(a) = PB(c)/SB(a) and Y2 = ID(d) = IB(d) for c

γ−−→ d , and Y1 = rad ID(b) =rad IB(b) and Y2 = SD(c) = SB(c) for c

γ←−− d . Consider the modified category Λ′ obtainedfrom Λ by splitting at the vertices b, c, d as follows


ω

σ2

σ1

ξ

c1

β1

γ1

C

b1 d1

δ2δ1

c2 = c

β2=β

γ2=γ

b2 = b

α2=α

d2 = d

a

and keeping the remaining parts of Λ unchanged. Observe that X0 and X1 are the unique inde-composable D-modules L such that HomD(X,L) �= 0 and Lα �= 0. Applying Lemma 4.4 again,we then conclude that the indecomposable Λ-modules

PΛ(ω) = X′0 = (K, X0,1), IΛ(a) = X′

1 = (K, X1,1),

U11 =(

K,X1 ⊕ Y1,

(11

)), U12 =

(K,X1 ⊕ Y2,

(11

)),

are the unique indecomposable Λ-modules N with Nξβα �= 0. Let XΛ be the full subcat-egory of indΛ′ formed by all modules except the six modules having support in the fullsubcategory of Λ′ given by b1, c1, d1, and YΛ be the full subcategory of indΛ formedby all modules except X′

0, X′1, U11 and U12. Then as above we define a canonical exact

functor FΛ : modΛ′ → modΛ which preserves the indecomposable modules, and whose re-striction to XΛ defines a dense functor XΛ → YΛ reflecting the isomorphisms. Observe alsothat the splitting Λ′ of Λ induces the corresponding splitting of the category Ω to the cate-gory Ω ′ which is a blowup of a D-extension of B of one of the types (d1) or (d2). Indeed,as in the former case, take U = τ 2

BY2. Then the support S(U) of HomB(U,−)|Γ is of theform

U = U0 U1 U2 U3 · · ·

with U1 = τBY1 and Un = Xn−2 for n � 2, and hence U can be the pivot of an admissible op-

eration of type (ad 1) or a D-extension of type (d1). For cγ

d, consider the D-extension

B ′ = B[U ] � H of B of type (d1), where H is the path algebra of the quiver
Kω


d1

ω c1

b1.

For cγ

d, consider first the coil enlargement B = (B × E)[U ⊕ P ] of B of type (ad 1)with pivot U , E the algebra of 2 × 2 upper triangular matrices, and P the unique indecompos-able projective–injective E-module, and then the D-extension B ′ = B[P ] of type (d2), with theparameter t = 0, creating the quiver

d1

ω c1

b1.

In the both cases, the blowup of B[X] inside Ω induces the corresponding blowup Ω ′ of B ′,containing the splitting Λ′ of Λ as a convex subcategory.

As in the case of D-extension of type (d3), we also show that the indecomposable Λ-modulesX′

0, X′1, U11 and U12 are the unique indecomposable A-modules N with Nξβα �= 0.

(3) We have also the dual splitting procedures which allow to modify all convex blowupsof D-coextensions Λ = [X]B of D-coil algebras B of types (d3∗) and (d4∗) inside A to thecorresponding blowups of coextensions of types (d1∗) or D-coextensions of types (d1∗) and (d2∗)of D-coil algebras.

A simple inspection of the frames of the hypercritical algebras (see [74]) shows that in thesesplitting procedures (for D-extensions and D-coextensions) we do not create convex hypercriticalsubcategories.

Therefore, all these local splitting replacements Λ′ of Λ modify the D-algebra A to a mildD-algebra A′. Further, the related functors FΛ : modΛ′ → modΛ described above extend (inthe obvious way) to an exact functor F : modA′ → modA which preserves the indecomposablemodules and defines a dense, reflecting isomorphisms, functor X → Y , where X is the cofinitefull subcategory of indA′ given by all modules except the modules from indΛ′ \ XΛ, and Y isthe cofinite full subcategory of indA given by all modules except the modules from indΛ \ YΛ,for all convex blowups Λ of D-extensions and D-coextensions of types (d3), (d4), (d3∗), (d4∗)inside A. Clearly, if A′ is tame then A is also tame. This finishes the proof. �

We associated above (in a canonical way) to any D-algebra A a mild D-algebra A′ whoserepresentation theory controls (up to finitely many isoclasses of indecomposable modules) therepresentation theory of A. The following example illustrates this procedure.

Example 4.6. Let A be the algebra given by the bound quiver


26

24

23 25

5

30 29

4 22

3′ 3′′ 6 28 21′

2

21′′

1 7 20 15

27

9 8

16

10

11

12′′ 14

12′ 17′′

13

17′

19 18

We show first that A is a D-algebra. Denote by A1 the bound quiver algebra obtained from A

by identifying the vertices 3′ = 3 = 3′′, 12′ = 12 = 12′′, 17′ = 17 = 17′′, 21′ = 21 = 21′′, andthe corresponding connected arrows. Clearly, A is the blowup of A1 at the vertices 3, 12, 17, 21.Let B be the convex subcategory of A1 given by all objects except 23, 24, 25, 26 and 30, and H

the convex subcategory of B given by the objects 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Observe that H isa critical algebra of type D9. Then, B is a D-coil algebra, obtained from the critical algebra H

by five admissible operations of type (ad 1), creating the sets of vertices {20}, {21}, {22}, {28},{29}, and eight admissible operations of type (ad 1∗), creating the sets of vertices {11}, {12},{13,14,15}, {16}, {17}, {18}, {19}, {27}. Further, the convex subcategory A2 of A1 given by allvertices except 24, 25, 26 and 30 is a D-extension B[X] of type (d4), creating the vertex 23and the arrows 23 −→ 22, 23 −→ 15, and the blowup Ω of A2 at the vertices 3, 12, 17, 21 isthe convex subcategory of A given by all objects except 24, 25, 26, 30. In particular, Ω is a D-algebra. Consider now the critical convex subcategory H1 of type D6 of A given by the vertices21′, 21′′, 22, 23, 14, 15, 16, and the convex subcategory B1 of A given by the vertices 21′,


21′′, 22, 23, 14, 15, 16, 24, 25. Observe that B1 is a coil enlargement of H1 by one admissibleoperation of type (ad 1), creating the vertices 24, 25. Hence the convex subcategory Σ of A givenby the vertices 21′, 21′′, 22, 23, 14, 15, 16, 24, 25, 26 is a D-extension of B1 of type (d2), creatingthe vertex 26. Finally, consider the critical convex subcategory H2 of type D8 of A given by thevertices 3′, 3′′, 4, 5, 6, 7, 8, 9, 10, and the convex subcategory B2 of A given by the vertices 3′, 3′′,4, 5, 6, 7, 8, 9, 10, 27, 28, 29. Then B2 is a coil enlargement of H2 by one admissible operationof type (ad 1∗), creating the vertex 27, and two admissible operations of type (ad 1), creating thevertices 28 and 29. Then the convex subcategory Θ of A given by the vertices 3′, 3′′, 4, 5, 6, 7,8, 9, 10, 27, 28, 29, 30 is a D-extension of B2 of type (d2). Finally, A is the pushout glueing

A = ((Ω �

H1B1) �

B1Σ

) �B2

Θ,

and consequently is a D-algebra. In the notation of the proof of Proposition 4.5, we may takeΩ = Λ = D[X], where D is the blowup of B at the vertices 3, 12, 17, 21 and X = radPΛ(23).The above considerations also show that the associated mild D-algebra A′ is obtained from A byonly one splitting, at the vertices 14, 15, 16. Therefore, A′ is of the form

26

24

23 25

5

30 29

4 22 151

3′ 3′′ 6 28 21′

2

21′′

1 7 20 141 161

27

9 8 15

10

11

12′′ 14 16

12′ 17′′

13

17′

19 18


We also note that, applying the degeneration procedures presented in Section 2, we are not ableto degenerate the convex subcategory of A given by the vertices 8, 11, 12′, 12′′, 13, 14, 15, 16,20, 21′, 21′′, 22 and 23 to a special biserial category, and consequently the replacement of A

by A′ is necessary (see Example 4.8).The second type of obstructions which do not allow to apply the degeneration results presented

in Section 2 to arbitrary D-algebras are created by some bad configurations of zero-relations. Forinstance, for the D-algebra A considered in Example 4.6, the presence of the zero-relation on thepath from the vertex 30 to the vertex 27 and absence of the zero-relation on the path from the ver-tex 29 to the vertex 27 do not allow apply Lemma 2.3 to the subcategory given by the vertices 28,29 and 30. In order to jump over such problems, we introduce the concept of a smooth D-algebra.

Let B be a D-coil algebra, Γ the large coil of ΓB , and X an indecomposable module in Γ .Assume that X is the pivot of an admissible operation of type (ad 1), (ad 2), or (ad 3). We say thatthe pivot X is maximal if Γ does not contain a pivot X′ of an admissible operation of the sametype (ad 1), (ad 2), or (ad 3), such that S(X) is a proper convex subcategory of S(X′). Observethat if X is the pivot of an admissible operation of type (ad 2) then X is maximal. Similarly, ifX is the pivot of a D-extension of B of type (d1) then X is said to be maximal provided X ismaximal as the pivot of an admissible operation of type (ad 1). Further, if X is the pivot of aD-extension of B of type (d2) then X is said to be maximal if t = 0 and Γ does not contain apivot X′ of a D-extension of B of type (d2) such that S(X) is a proper convex subcategory ofS(X′). We also note that, if X is the pivot of a D-extension of B of type (d3) or (d4), then X ismaximal, that is S(X) is not a proper convex subcategory of S(X′) for a pivot X′ ∈ Γ of a D-extension of B of type (d3) or (d4). Dually, one defines maximal copivots of the dual operations(ad 1∗), (ad 2∗), (ad 3∗), (d1∗), (d2∗), (d3∗), (d4∗).

A D-coil algebra B is said to be smooth if B is a coil enlargement of a critical algebra C oftype Dn invoking only admissible operations with maximal pivots and maximal copivots. A D-extension (respectively, D-coextension) A of a D-coil algebra B is said to be smooth providedthe pivot of the D-extension operation (respectively, the copivot of the D-coextension operation)is maximal. Finally, a D-algebra A is said to be smooth if all D-coil algebras, D-extensions andD-coextensions, occurring in the recursive definition of A, are smooth.

Proposition 4.7. Let A be a D-algebra. Then there is a smooth D-algebra A# such that A is afactor algebra of A#. In particular, if A# is tame then A is tame.

Proof. We will present a canonical procedure leading from the D-algebra A to a smooth D-algebra A#. Clearly, if A is a smooth D-algebra, we set A# = A. Assume it is not the case. We willreplace inductively all nonmaximal admissible operations, D-extensions and D-coextensions,occurring in the recursive procedure from a convex critical subcategory C of A to A, by thecorresponding maximal operations. We divide the proof into several steps.

(1) Assume that A admits a convex subcategory Ω which is a blowup of a nonmaximal en-largement Λ of a D-coil algebra B by an admissible operation of type (ad 1), say with the pivot X.Then the large coil Γ of ΓB contains a maximal sectional path

X′ = X′0 −→ X′

1 −→ · · · −→ X′m −→ X1 −→ X2 −→ · · ·

with X′m = X = X0 for some m � 1, S(X′) is the path category of this sectional path and S(X)

is the path category of the infinite subpath starting from X = X′m. In particular, X′ is the pivot

of an admissible operation of type (ad 1). Invoking now the structure result Proposition 3.1, weconclude that Λ admits a convex subcategory of the form


ω = a0α0

a1α1 · · · ar−1

αr−1ar · · ·bs

Cβs

bs−1βs−1 · · · b1

β1b0

where ω is the extension vertex of the one-point extension of the admissible operation of type(ad 1) with the pivot X, a1, . . . , ar−1 are vertices of an extension branch of the convex sub-category B+ (if r � 2), b0, b1, . . . , bs−1 are vertices of a coextension branch of the convexsubcategory B−, r � 1, s � 1, C is a critical convex subcategory of B or the vertex ar = bs ,and we have a zero-relation α0α1 · · ·αr−1uβs · · ·βp = 0 in Λ for some p ∈ {1, . . . , s}, with u asubpath of C from ar to bs (possibly the trivial path at ar = bs ). Replacing the (ad 1)-operationwith the pivot X by the (ad 1)-operation with the pivot X′ (with the same upper triangular matrixalgebra) is equivalent to removing the zero-relation from a0 to bp−1. Then the blowup Ω of Λ

can be replaced by the corresponding blowup Ω ′ of Λ′, obtained by removing the zero-relationfrom a0 to bp−1 in Ω .

(2) Assume now that A admits a convex subcategory Ω which is a blowup of a nonmaximalone-point extension Λ = B[X] of type (ad 3) of a D-coil algebra B . Then the large coil Γ of ΓB

contains a full translation subquiver of the form

Y ′1 Y ′

2 · · · Y ′m+1 Y2 · · · Yt

X′ = X′0 X′

1 · · · X′m X1 · · · Xt−1 Xt Xt+1 · · ·

with X′m = X = X0 and Y ′

m+1 = Y1 for some m � 1, S(X′) is the bound quiver category of thisquiver bound by the mesh relations of the squares, and S(X) is the convex subcategory of S(X′)given by all successors of X = X′

m in S(X′). Applying Proposition 3.1, we conclude that B[X]admits a convex subcategory of the form

a0 = ωα0

γ

a1α1

...

ar−1αr−1

ar

...C c0

σ

δ0

bs

bs

c1δ1

bs−1

βs−1..

.

... δn−1

cn

b1

β1 b0


where ω is the extension vertex of B[X], Xt−1 is the injective module IB(b0), a1, . . . , ar−1 arevertices of an extension branch of the convex subcategory B+ (if r � 2), b0, . . . , bs−1, c0, c1, . . . ,

cn are vertices of a coextension branch of B−, C is a critical convex subcategory of B or thevertex ar = bs , r, s, n � 1, γ σ = α0 · · ·αr−1uβs · · ·β1 for a subpath u of C from ar to bs (pos-sibly the trivial path at ar = bs ), γ δ1δ2 · · · δp = 0 for some p ∈ {0, . . . , n − 1}, and δ0 · · · δn−1is the maximal subpath of QB starting with the arrow δ0. Moreover, the indecomposable B-module X (respectively, X′), considered as a representation of the bound quiver of B , hasone-dimensional vector spaces at the vertices a1, . . . , ar , b0, . . . , bs, c0, . . . , cp (respectively,a1, . . . , ar , b0, . . . , bs, c0, . . . , cn). Therefore, replacing the (ad 3)-pivot X by the maximal (ad 3)-pivot X′ is equivalent to removing the zero-relation γ δ1 · · · δp = 0. Clearly, then the blowup Ω

of Λ can be replaced by the blowup Ω ′ of Λ′ = B[X′] obtained by deleting the zero-relation ona path from ω to cp+1.

The replacement procedure for a blowup of a nonmaximal D-extension of a D-coil algebra oftype (d1) inside A is the same as for the blowup of the corresponding admissible operation oftype (ad 1), induced by the pivot of the operation (d1).

(3) Finally, assume that A admits a convex subcategory Ω which is a blowup of a nonmaximalone-point extension Λ = B[X] of type (d2) of a D-coil algebra B .

(3.1) Assume first that the large coil Γ of ΓB contains a full translation subquiver of the form

Y1 Y2 · · ·

X′ = X′0 X′

1 · · · X′m X1 · · · Xt Xt+1 · · ·

with X′m = X = X0 for some m � 0, t � 0, with m + t � 1, S(X′) is the bound quiver category

of this quiver bound by the mesh relations of the squares, and S(X) is the convex subcategoryof S(X′) given by all successors of X = X′

m in S(X′). Moreover, Y1 is projective and τBY1is injective. We may assume that X′ is the maximal module in the coil Γ with this property.Applying Proposition 3.1, we infer that Λ = B[X] contains a convex subcategory of the form

ωγ

a1α1

a2 · · · arαr

ar+1 · · ·bs

βs

C

bs−1βs−1 · · · b1

β1b0

a0α0

where ω is the extension vertex of B[X], a0, a1, . . . , ar are vertices of an extension branch ofthe convex subcategory B+, b0, b1, . . . , bs−1 are vertices of a coextension branch of B−, C is acritical convex subcategory of B or C consists of the vertex ar+1 = bs , r � 1, s � 1, B[X] hasa zero-relation α0α1 · · ·αruβs · · ·βp = 0 for some p ∈ {1, . . . , s}, and possibly a zero-relationγ α1 · · ·αruβs · · ·βq = 0 for some q ∈ {1, . . . , p}, for a subpath u of C (possibly the trivial pathat ar+1 = bs ). Moreover, Y1 is the projective module PB(a0) and τBY1 is the injective moduleIB(bp−1). Let B1 be the convex subcategory of B given by all objects except a0. Then B1 isa D-coil algebra such that B = B1[Xt ] is obtained from B1 by the one-point extension of type(ad 1) with the pivot Xt . Consider now the D-coil algebra B ′ = B1[X′] obtained from B1 by theone-point extension of type (ad 1) with the pivot X′. Then the large coil Γ of ΓB is modified to


a large coil Γ ′ of ΓB ′ such that the support S(X′) of the functor HomB ′(U,−)|Γ ′ is the boundquiver category of the quiver of the form

Y ′1 Y ′

2 · · · Y ′m+1 Y ′

m+2 · · · Y ′m+t+1 Y ′

m+t+2 · · ·

X′ = X′0 X′

1 · · · X′m X1 · · · Xt Xt+1 · · ·

Hence, the one-point extension Λ′ = B ′[X′] is a D-extension of type (d2) of the D-coil alge-bra B ′. Moreover, the maximality of X′ implies that it is a maximal D-extension of B ′. Further,the quiver QΛ′ of Λ′ coincides with the quiver QΛ of Λ, a0 is the extension vertex of the one-point extension B ′ = B1[X′], ω is the extension vertex of the D-extension Λ′ = B ′[X′], and Λ′is obtained from Λ by removing the zero-relation α0α1 · · ·αrβs · · ·βp = 0, and the zero-relationγ α1 · · ·αruβs · · ·βq = 0, if such a zero-relation exists.

(3.2) Assume now that the large coil Γ of ΓB contains a full translation subquiver of the form

Y ′1 Y ′

2 · · · Y ′m Y1 Y2 Y3 · · ·

X′ = X′0 X′

1 · · · X′m−1 X0 X1 X2 · · ·

where X0 = X, m � 1, S(X′) is the bound quiver category of this quiver bound by the meshrelations of the squares, and S(X) is the convex subcategory of S(X′) given by all successors ofX = X0 in S(X). We may assume that X′ is the maximal module in the coil Γ with this property.Applying Proposition 3.1 again, we conclude that Λ = B[X] contains a convex subcategory ofthe form

ωγ

a1α1

a2 · · · arαr

ar+1 · · ·bs

βsC

bs−1βs−1 · · · b1

β1b0

a0α0

where ω is the extension vertex of B[X], r � 0, s � 1, C is a critical convex subcate-gory of B or C consists of the vertex ar+1 = bs , a0, a1, . . . , ar are vertices of an extensionbranch of the convex subcategory B+, or r = 0 and a0, a1 belong to C, b0, b1, . . . , bs−1are vertices of a coextension branch of the convex subcategory B−, Λ = B[X] has azero-relation γ α1 · · ·αruβs · · ·βp = 0, for some p ∈ {1, . . . , s}, and possibly a zero-relationα0α1 · · ·αruβs · · ·βq = 0, for some q ∈ {1, . . . , p}, if the edge a0 — a1 is the arrow a0 → a1,for a subpath u of C (possibly the trivial path at ar+1 = bs ). Replacing the one-point ex-tension B[X] by the one-point extension B[X′] is equivalent to removing the zero-relationγ α1 · · ·αruβs · · ·βp = 0, and adding the zero-relation γ α1 · · ·αruβs · · ·βq = 0 if the edgea0 — a1 is the arrow a0 → a1 and the relation α0α1 · · ·αruβs · · ·βq = 0 exists.

In both cases, the blowup Ω of Λ = B[X] can be replaced by the corresponding blowup Ω ′of Λ′ = B ′[X′] or Λ′ = B[X′], obtained from Ω by deleting the obvious zero-relations cor-


responding to the zero-relations removed in the replacement of Λ by Λ′. Therefore, applyingthe above replacements for the blowups of all nonmaximal admissible operations, D-extensionsor D-coextensions, involved in the definition of A, we reach a smooth D-algebra A# (uniquelydetermined by A). Obviously A is a factor algebra of A#. This finishes the proof. �Example 4.8. Let A be the algebra given by the bound quiver

32

31

11 30

28

29

5 26 27

2 4 6 23 35 40 38

34′

24 34′′

3 12 7 22 33 37

9 25

1 8 21 36

10

13 20

14′′ 19′

14′ 19′′

15 18 39

16′′

16′

17


We show first that A is a D-algebra. Denote by A1 the bound quiver algebra obtained from A

by identifying 14′ = 14 = 14′′, 16′ = 16 = 16′′, 19′ = 19 = 19′′, 34′ = 34 = 34′′, and the corre-sponding connected arrows. Then A is the blowup of A1 at the vertices 14, 16, 19, 34. Let B bethe convex subcategory of A1 given by all objects except 11, 29, 30, 31, 32 and 40, and H bethe convex subcategory of B given by the objects 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Observe that H

is a critical algebra of type D9. We claim that B is a coil enlargement of H , and hence is a D-coil algebra. Indeed, B is obtained from H by the following sequences of admissible operations:eight admissible operations of type (ad 1∗), creating the sets of vertices {12}, {13}, {14}, {15},{16}, {17,18,19,20,21,22,23}, {24}, {25}, one admissible operation of type (ad 2), creating thevertex 26, seven admissible operations of type (ad 1), creating the sets of vertices {27}, {28},{33}, {34}, {35,36}, {37}, {38}, and one admissible operation of type (ad 3∗), creating the ver-tex 39. Note that the admissible operations of type (ad 1), creating the vertices 27 and 28, and theadmissible operation of type (ad 3∗), creating the vertex 39, are not maximal. Consider now theD-extension D1 of B of type (d2), creating the vertex 11, the D-extension D2 of B of type (d2),creating the vertex 29, and the D-extension D3 of B of type (d2), creating the vertex 40. Thenthe pushout glueing D = (D1 �

BD2)�

BD3 is a D-algebra, whose blowup Σ at the vertices 14, 16,

19, 34 is the convex subcategory of A given by all objects except 30, 31, 32. We note that theD-extension D1 of B is not maximal. Denote by Θ the convex subcategory of A given by thevertices 1, 2, 3, 4, 5, 6, 7, 8, 23, 26, 27, 28, 29, 30, 31 and 32, by C the convex subcategoryof Θ given by all objects except 30, 31, 32, and by E the convex subcategory of Θ given by allobjects except 32. Then C is a critical algebra of type D12 and E is a D-coil algebra obtainedfrom C by one admissible operation of type (ad 1), creating the vertices 30 and 31. Further, Θ isa D-extension of the algebra E of type (d2), creating the vertex 32. Finally, A is the pushoutglueing

A = Σ �C

Θ,

and consequently is a D-algebra. Moreover, the associated smooth algebra A# is obtained from A

be removing the four zero-relations, namely the zero-relations on the paths 11 −→ 5 −→6 −→ 12, 27 −→ 23 −→ 24 −→ 25, 28 −→ 27 −→ 23 −→ 24, and 38 −→ 37 −→ 36 −→ 39.

Combining the procedures presented in the proofs of Propositions 4.5 and 4.7, we mayassociate (in a canonical way) to an arbitrary D-algebra A the mild and smooth D-algebraA∗ = (A′)# = (A#)′. Moreover, if A∗ is tame then A is also tame. The following propositioncompletes the proof of Theorem 4.3.

Proposition 4.9. Let A be a mild and smooth D-algebra. Then A degenerates to a special biserialalgebra. In particular, A is tame.

Proof. This is done in several steps by iterated applications of the degeneration procedures de-scribed in Section 2 to the convex subcategories of A which are not special biserial. We dividethe proof into several steps.

(1) We degenerate first the convex subcategories of A created by blowups to the correspondingspecial biserial algebras subcategories. Assume that A contains a convex subcategory given bythe quiver


a1β1

x

α1

α2

y

a2β2

bound by α1β1 = α2β2, created by the blowup at a vertex a. It follows from the definition ofthe blowup that the conditions of Lemma 2.4 (for the scalar μ = −1) are satisfied, and hence wemay degenerate A to an algebra where this blowup is replaced by the convex subcategory of theform

xα

y

ε

βz

with αβ = 0, ε2 = 0, and the corresponding modifications of all relations of A invoking the ar-rows α1, α2, β1, β2. Applying the above procedure, we degenerate the algebra A to an algebra A1obtained from A by replacing all blowups inside A by the corresponding special biserial convexsubcategories. We also note that in this degeneration of A to A1, our additional assumptions thatA is mild and smooth are not applied.

(2) In the second step, we degenerate the nonspecial biserial convex subcategories of A1,created by the admissible operations in the D-coil algebras involved in the recursive definitionof the D-algebra A, to the corresponding special biserial convex subcategories. Recall that, byProposition 3.1, every D-coil algebra B is a suitable glueing of a convex tubular extension B+(obtained from a critical convex subcategory C of type Dn by an iterated application of admis-sible operations (ad 1)) and a tubular coextension B− (obtained from the same convex criticalsubcategory C by an iterated application of admissible operations (ad 1∗)), and the glueing rela-tions are determined by the pivots and copivots applied in the coil enlargement of C to B . Thepivots and the copivots of the admissible operations are completely determined by the simple reg-ular modules applied in the one-point extensions and the one-point coextensions of C inside B ,and the admissible operations done so far. Moreover, the simple regular modules over the criti-cal algebras of types Dn are completely described in [45, Section 2]. Invoking this description,we conclude that applying the admissible operations of types (ad 1) and (ad 1∗), we may createnonspecial biserial convex subcategories only of the following forms

•α γ

•

β

•,

σ

•

•α

•β

•γ

•,

•

•α

•β

•

•γ

•αβ + γ σ = 0

(equivalently αβ = γ σ)αβγ = 0 αβγ = 0


(compare the pg-critical algebras of types (5), (7), (10), and their duals). Here, again the assump-tions of Lemmas 2.4 and 2.6 are satisfied, and hence we may degenerate A1 to an algebra A2

where all these convex subcategories are replaced, respectively, by convex subcategories of theforms

•α

•β

ε

•

•α

•β

δ •γ

•

•

•α

σ •

β

•

•γ

•αβ = 0, ε2 = 0 αδ = 0, βγ = 0 αβ = 0, σγ = 0.

The admissible operations of types (ad 2) and (ad 2∗) in the D-coil algebras B , involvedin the definition of A, create the commutativity relations α1α2 · · ·αp = β1β2 · · ·βq , p,q � 2from the top vertex to the socle vertex of the created indecomposable projective–injective B-module. We note that one of the parallel paths α1α2 · · ·αp or β1β2 · · ·βq may contain (one ortwo) subpaths of length 2 involved into commutativity relations of length 2 created by one-pointextensions or one-point coextensions of the critical convex subcategory C of B by simple reg-ular C-modules. Moreover, in the D-algebra A, the blowups at the vertices of the both pathsα1α2 · · ·αp and β1β2 · · ·βq may occur. After applying the degenerations leading from A to A2

(described above), the commutativity relation α1α2 · · ·αp = β1β2 · · ·βq of the D-coil algebra B

is replaced in A2 by another commutativity relation γ1γ2 · · ·γr = σ1σ2 · · ·σs with r � p, s � q ,and possibly some of the arrows γ2, . . . , γr−1, σ2, . . . , σs−1 are loops. Clearly, if γi , for somei ∈ {2, . . . , r − 1} (respectively, σj , for some j ∈ {2, . . . , s − 1}) is a loop, then we have in A2

the relations γ 2i = 0, γi−1γi+1 = 0 (respectively, σ 2

j = 0, σj−1σj+1 = 0). Therefore, we keepthe commutativity relations γ1γ2 · · ·γr = σ1σ2 · · ·σs unchanged (in the further degenerationsof A2).

Assume now that, in the recursive definition of A, occurs a D-coil algebra B which isa coil enlargement of a critical algebra C using an admissible operation of type (ad 3) or(ad 3∗). Then we have in B a commutativity relation αβ = γ1γ2 · · ·γm, m � 2, given by aone-point extension of type (ad 3) or a one-point coextension of type (ad 3∗). As above, thepath γ1γ2 · · ·γm may contain (one or two) subpaths of length 2 involved into commutativityrelations of length 2 created by one-point extensions or coextensions of C inside B by sim-ple regular C-modules. Moreover, in the D-algebra A, the blowups at the vertices of the pathγ1γ2 · · ·γm may occur. Applying the degenerations leading from A to A2, the path γ1γ2 · · ·γm

of B is replaced in A2 by a path ξ1ξ2 · · · ξn, n � m, and A2 admits a convex subcategory of theform


• ξ2 • · · · • ξn−1 •ξn

•

ξ1

α

•

•β

σ

•

δ

•

with αβ = ξ1ξ2 · · · ξn, possibly only one of the arrows δ or σ occurs, and some of the arrowsξ2, . . . , ξn−1 are loops. Moreover, if both arrows δ and σ occur, then δσ = 0. Further, if ξi ,for some i ∈ {2, . . . , n − 1}, is a loop, then ξ2

i = 0 and ξi−1ξi+1 = 0. Finally, in the remain-ing relations of A2, the arrows α and β may occur only in zero-relations. Therefore, applyingLemma 2.5 (for the scalar μ = −1), we may degenerate A2 to an algebra with the same quiver butthe commutativity relation αβ = ξ1ξ2 · · · ξn replaced by the zero-relation αβ = 0. Therefore, wemay degenerate A2 to an algebra A3, where all commutativity relations αβ = ξ1ξ2 · · · ξn in A2,created by all admissible operations of types (ad 3) and (ad 3∗) applied in the D-coil algebrasinvolved in the definition of A, are replaced by the zero-relations αβ = 0.

In the final step, we degenerate the nonspecial biserial convex subcategories of A3 createdby the critical convex subcategories as well as the D-extensions and the D-coextensions of D-coil algebras, involved in the recursive definition of A, to the corresponding special biserialconvex subcategories. This is done by local application of Proposition 2.9. Namely, because A

is mild, only D-extensions of types (d1) and (d2) (respectively, D-coextensions of types (d1∗)and (d2∗)) may occur. Moreover, by Proposition 3.1 and the description of simple regular mod-ules and indecomposable regular modules of regular length 2 given in [45, Section 2], anyD-extension of type (d1) or (d2) (respectively, D-coextension of type (d1∗) or (d2∗)) of a D-coil algebra B creates a pg-critical convex subcategory Λ of this D-extension (respectively,D-coextension) of B . This pg-critical category Λ may be enlarged in A by some blowups,which in the degeneration process from A to A3 are replaced by the corresponding special bis-erial configurations of zero-relations. Further, because A is smooth, all applied D-extensionsand D-coextensions are maximal, and consequently the possible zero-relations obstructions(in arbitrary D-algebras) for applications of the degeneration Lemmas 2.3–2.8 are removed.Hence, as in the proof of Proposition 2.9, we may degenerate further in A3 the degenerationsof all blowups of the pg-critical categories occurring in A to special biserial convex subcate-gories. Therefore, A3, and hence A, degenerates to a special biserial algebra. This finishes theproof. �

We note that we constructed, in fact, a canonical degeneration of a mild and smooth D-algebra A to a special biserial algebra A.

Example 4.10. Let A be the D-algebra from Example 4.2. Then A is a mild and smooth D-algebra and degenerates to the special biserial algebra A given by the quiver


13

11 12 5 20

10−− 6

19α

σ1−− 3 4 7 18

β

14 8

γ

17

δ

ε−−

15 9

16

bound by the commutativity relation αβγ = σεδ and the zero-relations denoted by the dashed

lines. Here, the dashed loop •−− means that its square is zero.

Example 4.11. Let A be the D-algebra from Example 4.6. The mild smooth D-algebra A∗ isobtained from A by removing the zero-relation on the path 30 −→ 28 −→ 7 −→ 27 and splittingat the vertices 14, 15, 16. Applying the degeneration procedures presented in Section 2, we maydegenerate A∗ to the special biserial algebra given by the bound quiver

26

24

5 23

4 29

−−22 151 25

3−− 6 28 21−− 141−−

2 7 20 15

1 27 8 16

9−−

11 14

12−−

13 17−−

19 18

where all dashed lines denote the zero-relations.


Example 4.12. Let A# be the smooth algebra of the D-algebra A considered in Example 4.8.Then A∗ = A#, and, applying the degeneration procedures presented in Section 2, we may de-generate A∗ to the special biserial algebra given by the bound quiver

11 32 31

5 26 30

28− − − 38

− − −

2 4 6 27 35

23 34− − − 37

3 12 7 24 22 33

8 9 − − − 25 21 36

1 13 20

14− − − 19− − −

15 18 39

16− − −

17

where, except the commutativity relation from 26 to 17, the remaining dashed lines denote thezero-relations.

5. Extremal algebras

An indecomposable module X over a triangular algebra A = KQ/I is said to be extremal ifits support suppX = {i ∈ Q0: X(i) �= 0} contains all extreme vertices (sinks and sources) of Q.A triangular algebra A is called extremal if there is an extremal indecomposable finite dimen-sional A-module. The extremal algebras were introduced in [9] (and called in [58] essentiallysincere algebras) as a natural generalization of sincere algebras but include many other exam-ples.


Example 5.1. Let A be the bound quiver algebra given by the quiver with relations

• •• · · · • •

• a1 •...

• · · · •

• as •• · · · • •

• •

Observe that A is a D-coextension of type (d1∗) of a pg-critical algebra of type (3), and hence is aD-algebra. Then A admits an indecomposable finite dimensional module whose support containsall vertices with the exception of a1, . . . , as , hence A is extremal (see [46, Section 6]). Moreover,A is not a sincere algebra.

Observe that a strongly simply connected algebra A is tame if and only if every convex sub-category B of A which is extremal is tame. The main result of [58], partially recalled as (1.5), isthe following theorem.

Theorem 5.2. Let A be a triangular algebra satisfying the following conditions:

(i) A is extremal and strongly simply connected;(ii) qA is weakly nonnegative;

(iii) A contains a convex subcategory which is either representation-infinite tilted of type Ep

(p = 6, 7 or 8) or a tubular algebra.

Then A is either a tilted algebra or a coil algebra.

Among other ingredients, the following simple lemma is important in the proof of the abovetheorem.

Splitting Lemma 5.3. Let A be a triangular algebra and B = B0, B1, . . . ,Bs = A a familyof convex subcategories of A such that, for each 0 � i � s, Bi+1 = Bi[Mi] or Bi+1 = [Mi]Bi

for some indecomposable Bi -module Mi . Assume that the category indB of indecomposableB-modules admits a splitting indB = P ∨ J , where P and J are full subcategories of indB

satisfying the following conditions:

(S1) HomB(J , P ) = 0;(S2) for each i such that Bi+1 = Bi[Mi], the restriction Mi |B belongs to the additive category

add J of J ;


(S3) for each i such that Bi+1 = [Mi]Bi , the restriction Mi |B belongs to the additive categoryadd P of P ;

(S4) there is an index i with Bi+1 = Bi[Mi] and Mi ∈ J and an index j with Bj+1 = [Mj ]Bj

and Mj ∈ P .

Then A is not extremal.

Proof. Let x1, . . . , xr (respectively, y1, . . . , yt ) be those vertices at the quiver Q of A beingsources (respectively, targets) or arrows with target (respectively, source) in B . For each i, denoteby B+

i (respectively, B−i ) the maximal convex subcategory of Bi not containing any y1, . . . , yt

(respectively, x1, . . . , xr ). Let Pi (respectively, Ji ) be the full subcategory of indB−i (respec-

tively, of indB+i ) consisting of modules X such that X|B ∈ add Pi (respectively, X|B ∈ add Ji ).

We claim that indBi = Pi ∨ Ji and HomBi(Ji , P ) = 0. The proof of the claim follows from

induction as in [55, p. 1022].We get that indA = Ps ∨ Js with HomA(Js , Ps) = 0, Ps consists of B+

s -modules and Js

consists of B−s -modules. Moreover, by (S4), B �= B+

s and B �= B−s . Let X ∈ Ps and let y be

a sink in Q which is a successor of y1. Since B+s is convex in A, then y is not in B+

s , henceX(y) = 0. That is, X is not extremal. Similarly, any module Y ∈ Js is not extremal. We concludethat A is not extremal. �

The following application of the Splitting Lemma will be also used in the proof of the MainTheorem.

Proposition 5.4. Let A be an algebra with a convex subcategory C which is a critical algebra.Consider the splitting decomposition indC = P ∨ J , where J is the preinjective componentof the Auslander–Reiten quiver ΓC of C. Let C = B0,B1, . . . ,Bs = A be a family of convexsubcategories of A such that, for each 0 � i � s, we have Bi+1 = Bi[Mi] or Bi+1 = [Mi]Bi forsome indecomposable Bi -module Mi . Assume the following:

(i) A is extremal and strongly simply connected;(ii) the Tits form qA is weakly nonnegative;

(iii) whenever Bi+1 = Bi[Mi] (respectively, Bi+1 = [Mi]Bi ) we have Mi |C ∈ add J (respec-tively, Mi |C ∈ add P );

(iv) for some 0 � j � s, Bj+1 = Bj [Mj ] and the restriction of Mj to C belongs to add J .

Then A is a tame tilted algebra. In particular, A does not contain a pg-critical algebra.

Proof. The Splitting Lemma 5.3 implies that for all i we have Bi+1 = Bi[Mi] and the restrictionsMi |C belong to add J . Then, for each 0 � i � s, there is a splitting indBi = P ∨ Ji , where eitherJi is a preinjective component or i + 1 = s, Bs = A is a tilted algebra and Js is a connectingcomponent of ΓBs . Indeed, a simple induction argument (see [55, Proposition 3.2]) shows thatin case 0 � i < s and Ji is a preinjective component of ΓBi

with a complete slice Σi , then Ji+1is a connecting component of ΓBi+1 with a complete slice. In case Ji+1 is not a preinjectivecomponent of ΓBi+1 and Mi+1 is defined then the Tits form qBi+2 is not weakly nonnegative(proceed as in [47, (2.5)], since there are orthogonal indecomposable modules X1, . . . ,X5 suchthat HomBi+1(Mi+1,Xj ) �= 0 for all j ). This is a contradiction showing that i + 1 = s and A is atilted algebra. Finally, by [36], a tilted algebra A with weakly nonnegative Tits form is tame. �


6. Proof of the Main Theorem

Let A be a strongly simply connected algebra, and assume that the Tits form qA of A isweakly nonnegative. We will prove that then A is a tame algebra. Let A = KQ/I be a boundquiver presentation of A. We identify A with a K-category whose class of objects in the set Q0of vertices of the quiver Q, and modA with the category repK(Q, I) of finite dimensional rep-resentations of the bound quiver (Q, I) over K . For a module M in modA, we denote by dimM

the dimension-vector (dimK M(i))i∈Q of M . The support suppM of a module M in modA isthe full subcategory of A given by all objects i ∈ Q0 with M(i) �= 0. Since A is strongly simplyconnected, the convex hull 〈suppM〉 of suppM inside A is a strongly simply connected category,and M is an extremal module over 〈suppM〉. Finally, observe that the Tits form qΛ of a convexsubcategory Λ of A is the restriction of qA to Λ, and consequently is also weakly nonnegative.

In order to prove that A is tame, it is enough to show that, for each dimension-vectord = (di) ∈ N

Q0 , there exists a finite number of K[X]-A-bimodules Mi , which are finitely gen-erated and free as left K[X]-modules, and all but a finite number of isomorphism classes ofindecomposable A-modules M with dimM = d are of the form K[X]/(X − λ) ⊗K[X] Mi forsome i and some λ ∈ K . Hence, A is tame if and only if the convex hull 〈d〉 of any dimension-vector d ∈ N

Q0 inside A is a tame algebra. Accordingly, we may assume that A is an extremalalgebra, that is, there is an indecomposable finite dimensional A-module M with 〈suppM〉 = A.We know from Proposition 1.4 that, if A does not contain a pg-critical convex subcategory, thenA is of polynomial growth, and hence is tame. Moreover, it follows from Proposition 4.1 andTheorem 4.3 that every D-algebra is a strongly simply connected tame algebra. Therefore thefollowing theorem completes the proof of our main theorem.

Theorem 6.1. Let A be a strongly simply connected algebra satisfying the following conditions:

(i) A is extremal.(ii) qA is weakly nonnegative.

(iii) A contains a pg-critical convex subcategory.

Then A is a D-algebra.

Proof. It follows from Proposition 1.4 and the assumptions (ii), (iii) that A is not of polynomialgrowth. Then, applying Theorem 1.5, we conclude that A does not contain a convex subcategorywhich is either a representation-infinite tilted algebra of type Ep , 6 � p � 8, or a tubular algebra.Further, by Corollary 1.6, every critical convex subcategory of A is of type Dm, m � 4, and hencebelongs to one of the four families of algebras presented at the beginning of Section 4. Moreover,it is known that the Tits form of a tubular extension (respectively, tubular coextension) B of acritical algebra C is weakly nonnegative if and only if B is a tubular algebra or a representation-infinite tilted algebra of Euclidean type (see [5, (4.2)], [60, (3.3)], and [63, Sections 4 and 5]).Therefore, we conclude that, if a convex subcategory B of A is a tubular extension (respectively,tubular coextension) of a critical algebra, then B is a representation-infinite tilted algebra oftype Dm, for some m � 4.

Let Λ be a maximal convex subcategory of A satisfying the following conditions:

(i) Λ is a D-algebra.(ii) Λ contains a convex pg-critical subcategory.


Here, maximal means with a maximal number of objects. Clearly, such a maximal convex subcat-egory Λ exists, because A contains a convex pg-critical subcategory, and the pg-critical algebrasare D-algebras. Our aim is to show that Λ = A, and consequently that A is a D-algebra.

In order to show that Λ = A, we argue by contradiction. Suppose that Λ �= A. Since A isstrongly simply connected and Λ is a convex subcategory of A, applying Proposition 1.2, weconclude that there is a sequence Λ = Λ0,Λ1, . . . ,Λm = A of convex subcategories of A suchthat, for each i ∈ {0, . . . ,m − 1}, Λi+1 is a one-point extension Λi[Mi] or a one-point coex-tension [Mi]Λi of Λi by an indecomposable Λi -module Mi . We may assume (without loss ofgenerality) that there exists an indecomposable Λ-module M such that Λ[M] is a convex subcat-egory of A. We will show that this leads to a contradiction either with maximality of Λ or withour assumption that A is extremal.

Denote by w the extension vertex of the one-point extension Λ[M]. Then w is a source ofthe quiver QΛ[M] of Λ[M], and there is in Q an arrow w → a with a in QΛ. Since Λ is aD-algebra, by Proposition 4.1, there is a D-coil algebra B which is a convex subcategory of Λ

and contains the object a. We note (see Example 4.2) that there are usually many convex D-coilalgebras inside Λ containing the fixed object a.

Let B be a fixed maximal D-coil algebra which is a convex subcategory of Λ and contains a.We denote by MB the restriction of the Λ-module M to B . Since B is a convex subcategory of Λ,MB can be considered as a Λ-module (by extending MB by zero vector spaces at the objects of Λ

which are not in B). Observe also that the one-point extension B[MB ] is a convex subcategoryof A, and so is strongly simply connected. Since B is a connected algebra, applying Proposi-tion 1.1, we then conclude that the B-module MB is indecomposable. By definition, the D-coilalgebra B is a coil enlargement of its unique critical convex subcategory C = CB (of type Dn)using only simple regular modules from a fixed stable tube of rank n−2 in the Auslander–Reitenquiver ΓC of C. Then, by general theory (see [5, Theorem 4.2]), the Auslander–Reiten quiver ΓB

of B is of the form

ΓB = PB ∨ CB ∨ QB

where PB = PB− is the preprojective component of the maximal tubular coextension B− of C

inside B , QB = QB+ is the preinjective component of the maximal tubular extension B+ of C

inside A, and CB is a P1(K)-family of pairwise orthogonal standard coils consisting of one (large)coil having at least n − 2 rays and at least n − 2 corays, two stable tubes of rank 2, and a familyof stable tubes of rank 1 indexed by K \ {0}. The ordering from the left to the right indicatesthat there are nonzero morphisms in modB only from any of these components to itself or to thecomponents on its right. We will analyze now the algebra structure of B[MB ], depending on theposition of the indecomposable B-module MB in ΓB . Therefore, we have three cases to consider.We abbreviate N = MB .

(I) Assume first that N belongs to the preprojective component PB = PB− . Since B− is atubular coextension of C inside the D-algebra Λ, B− is a representation-infinite tilted algebraof type Dm, m � n, obtained from C by the pushout glueing of C with the branches at thecoextension vertices of the applied one-point coextensions of C by (pairwise nonisomorphic)simple regular modules from a fixed stable tube of rank n − 2 in ΓC (see the dual results to thosein [63, (4.9)]). Further, all modules from the preprojective component PC of C lie in PB− . Infact, the restriction of any indecomposable module from PB− to C is either zero or a direct sumof indecomposable modules from PC . We claim now that the restriction R of the module N to C

is zero. Suppose R is nonzero. Since C[R] is a convex subcategory of Λ, C[R] is strongly simply


connected, and hence R is an indecomposable (preprojective) C-module. Then it is well knownthat the Tits form of the one-point extension C[R] is not weakly nonnegative (see [47, (2.5)],[62, (2.5)]). On the other hand, C[R] is a convex subcategory of A, and hence, by our assump-tion (ii), the Tits form of C[R] is weakly nonnegative. Therefore, indeed the restriction of N

to C is zero, and N is an indecomposable representation of a branch L of B−, connected to C

by the coextension vertex of a one-point coextension of C by a simple regular C-module. Fur-ther, the bound quiver algebras of branches are strongly simply connected representation-finitespecial biserial algebras, so the support of N is the path algebra of a connected linear quiver,and N has the one-dimensional vector space at each vertex of its support. Moreover, by our as-sumption, N belongs to the preprojective component PB− of ΓB− , and consequently we haveHomB−(D(B−),N) = 0, for the injective cogenerator D(B−) in modB−. Then we concludethat B contains a convex subcategory Σ of the form

C

•v

• · · · • •a1

•a2

· · · •ar−1

•ar

•ar+1

· · · •as−1

•as

where v is the coextension vertex of a one-point coextension of the critical algebra C by a simpleregular C-module (possibly there is only one arrow connecting C and v), suppN is the categorygiven by the objects a1, . . . , ar , . . . , as , 1 � r � s, and possibly v = a1. Since the one-pointextension B−[N ] is a convex subcategory of A, B−[N ] is a strongly simply connected algebrawith weakly nonnegative Tits form. Applying Proposition 1.3, we then conclude that B−[N ]does not contain a convex hypercritical subcategory. In particular, the one-point extension Σ[N ]does not contain a convex hypercritical subcategory. We also note that Σ is a convex subcategoryof a pg-critical algebra (see Section 1), containing only one critical algebra, namely the criticalalgebra C. Invoking the shapes of hypercritical algebras (see [74]), we deduce that the supportof N is given by one of the quivers

(α) •a1

•a2

· · · •as−1

•as

(r = 1, s � r),

(β) •a1

•a2

· · · •as−2

•as−1

•as

(r = 1, s � 3),

(γ ) •a1

•a2

· · · •at

•at+1

· · · •as−1

•as

(r = 1, t � r, s � t + 2).

Assume that suppN is of the form (α). Then B−[N ] contains the convex subcategory Σ[N ]of the form

C

•ξ

w

• • · · · • • • · · · •
v a1 a2 as


bound only by the relations in Σ , and w is the extension vertex of B−[N ] (equivalently, Σ[N ]).Further, there is possibly an arrow u

η−→ as in the branch L, with u not in Σ . Moreover, if L

admits an arrow as�−→ b, with b not in Σ , then ξ� = 0 (because b is not in suppN ) and clearly

η� = 0, if an arrow uη−→ as in L exists (the relation from L). Observe that, if L does not admit

an arrow uη−→ as (with u not in Σ ), then B−[N ] is a tubular coextension of C, with the branch L

enlarged by one arrow wξ−→ as . Observe also that then B[N ] is a D-coil algebra, obtained from

the tubular coextension B−[N ] of C by the sequence of admissible operations of types (ad 1),(ad 2), (ad 3) leading from B− to B . Assume now that L admits an arrow u

η−→ as with u not in Σ .Then it follows from [46] that the convex subcategory Θ of B−[N ] given by the objects of Σ

and the objects u, w is a pg-critical algebra. In particular, the vertices u,w,as, . . . , a2, a1, . . . , v

belong to a unique critical convex subcategory C′ of Θ (different from C). We also note that,since B−[N ] does not contain a hypercritical convex subcategory, η is the unique arrow of thebranch L attached to the vertex u. We claim that the one-point extension B[N ] is a D-algebra.Observe first that the convex subcategory D of B−[N ] given by the objects of C, L and thenew extension object w is a D-algebra, obtained from the critical algebra C′ by a sequence ofadmissible operations of type (ad 1∗), creating the vertices of L which are not in Σ , a sequence ofadmissible operations of type (ad 1∗), creating some vertices of C which are not in C′, and finallyone D-extension of type (d1) or (d2), creating the remaining part of C. Consider also the convexsubcategory E of D given by all objects of D except w. Note that E is a tubular coextension of C

and so is a D-coil algebra. Then E is a convex subcategory of the D-coil algebra B , and B[N ]is the pushout glueing B[N ] = B �

ED of B and D along E. In particular, B[N ] is a D-algebra.

Finally, we note that, in the both cases, B[N ] is a convex subcategory of Λ[M], and hence of A.Assume that suppN is of the form (β). Then B−[N ] contains a convex subcategory of the

form

C

•ξ

w

•v

• · · · • •a1

•a2

· · · •as−2

• η

as−1

•as

bound only by the relations in Σ , and w is again the extension vertex of B−[N ]. Further, if L

admits and arrow as�−→ b, then ξη� = 0 in B−[N ]. It follows also from [46] that Θ = Σ[N ] is

a pg-critical algebra, and consequently the vertices w,as, . . . , a2, a1, . . . , v belong to a uniquecritical convex subcategory C′ of Θ (different from C). Since B−[N ] does not contain a hy-percritical convex subcategory, we conclude that the branch L has neither an arrow u

σ−→ as−1nor an arrow x

γ−→ as with x �= as−1. We claim that B[N ] is a D-algebra. Observe first that theconvex subcategory D of B−[N ] given by the objects of C, L and w is a D-algebra, obtainedfrom the critical algebra C′ by a sequence of admissible operations of type (ad 1∗), creating thevertices of L which are not in Σ and some vertices of C which are not in C′, and finally oneD-extension of type (d1) or (d2), creating the remaining part of C. Consider now the convex sub-category E of D given by all objects of D except w. Then E is a tubular coextension of C, andhence E is a D-coil algebra. Moreover, E is a convex subcategory of the D-coil algebra B , andso B[N ] = B �

ED is a D-algebra. Finally, observe that B[N ] is a convex subcategory of Λ[M],

and hence of A.


Assume that suppN is of the form (γ ). Then B−[N ] contains a convex subcategory of theform

•ξ η

w

C

•v

• · · · •a1

•a2

· · · •γtat

• · · ·γt+1

• •γs−1 as

with t � 1, bound only by the relations in Σ and the commutativity relation ξγt = ηγs−1 · · ·γt+1

from the extension vertex w to at+1. Further, if L admits and arrow as�−→ b, then η� = 0 in

B−[N ]. Again Θ = Σ[N ] is isomorphic to a pg-critical algebra (see Section 1), and con-sequently the vertices w,as, as−1, . . . , a2, a1, . . . , v belong to a unique critical convex sub-category C′ of Θ (different from C). Finally, since B−[N ] does not contain a hypercriticalconvex subcategory, we conclude that the branch L has neither an arrow u

σ−→ as nor an ar-

row xβ−→ at (if t � 2). We claim that B[N ] is a D-algebra. The convex subcategory D of

B−[N ] given by the objects of C, L and w is a D-algebra, obtained from the critical alge-bra C′ by a sequence of admissible operations of type (ad 1∗), creating the vertices of L whichare not in Σ and some vertices of C which are not in C′, and one D-extension of type (d1)or (d2), creating the remaining part of C. Take now the convex subcategory E of D givenby all objects of D except w. Observe that E is a tubular coextension of C, and hence E

is a D-coil algebra. Moreover, E is a convex subcategory of the D-coil algebra B , and henceB[N ] = B �

ED is a D-algebra. Observe also that B[N ] is a convex subcategory of Λ[M], and

hence of A.Summing up, in the three considered cases, Λ[M] admits a convex subcategory B[N ], with

N = MB the restriction of M to B , which is moreover a D-algebra. It follows also from Propo-sition 1.2 that Λ[M] can be obtained from B[N ] by a sequence of one-point extensions andone-point coextensions by indecomposable modules. We will show now that Λ[M] is a D-algebra. This will lead to a contradiction with the maximality of Λ inside A, because Λ[M]is a convex subcategory of A.

Observe first that, if M = MB = N , then Λ[M] = Λ[N ] is the pushout Λ[N ] = Λ�B

B[N ],and consequently Λ[M] is a D-algebra.

Assume now that M �= MB . Let b be a vertex with M(b) �= 0 = N(b). Moreover we canchoose b such that there is a convex subcategory B ′ of Λ whose quiver has vertices in QB ∪ {b}.Since B ′ is strongly simply connected then B ′ = [L]B with L an indecomposable B-moduleand b the coextension vertex. Let us show first that there is an arrow ai → b. Otherwise thereis an arrow w → b. Recall that, by maximality of B , the algebra B ′ is not a coil algebra. Hencethere is a critical convex subcategory C′ of B ′ containing w. Consider a vector 0 �= x in theGrothendieck group of C′ with qC′(x) = 0. We get q[L]C′(2x + eb) < 0, where eb is the vectorwith value 1 at b and 0 everywhere else, a contradiction since [L]C′ is a convex subcategoryof A.

As before we consider three cases (α), (β), (γ ), depending on the shape of suppN .


(α) The algebra B ′ has the shape

C

• b •ξ

w

•v

• · · · • •a1

•a2

· · · •ai

· · · •as

bound only by the relations in Σ . Since the Tits form qB ′ is weakly nonnegative, then [47] impliesthat i = s. As observed in the first consideration of case (α), the composition w → as → b

vanishes, which contradicts that M(b) �= 0.(β) The algebra B ′ has the shape

C

• b • w

•v

• · · · • •a1

•a2

· · · •ai

· · · •as−1

•as

bound only by the relations in Σ . Clearly, B ′ contains a convex subcategory which is hereditaryof wild type, in particular the Tits form qB ′ is not weakly nonnegative, a contradiction.

(γ ) As in case (β) we get that the Tits form qB ′ is not weakly nonnegative, a contradictioncompleting the proof of case (I).

(II) Suppose that N ∈ CB . Let T be the tubular family in ΓC . We consider first the caseN ′ := NC �= 0. Since C[N ′] is convex in B[N ] which is strongly simply connected, then N ′ isindecomposable in T . We distinguish two cases.

(II.1) N ′ is a simple regular C-module. Hence C[N ′] is tilted of type Dn+1. Assume that B

is a coil enlargement of C using modules N1, . . . ,Ns in T as pivots or copivots. We distinguishseveral possible situations:

(i) N ′ = N1. Consider B1 the maximal coil enlargement of C by N1 and B the maximal coilenlargement of C by N2, . . . ,Ns . Then B1[N ′] is a pg-critical algebra and the pushout B[N ′] =B �

B1B1[N ′] is also a D-algebra. We shall prove that Λ[M] is a D-algebra which contradicts the

maximality of Λ.In fact, if N ′ = M then Λ[M] = Λ�

BB[N ′] is a D-algebra. Hence we may assume that

N ′ �= M . Let b be a vertex not in C such that M(b) �= 0. We may assume that the algebraE := [L]C[N ′][N ′] is convex in A, where L is the restriction to C[N ′][N ′] of the injective A-module Ib at the vertex b.

We shall show that the algebra E and therefore the pushout [L]B[N ′] = B[N ′] �C[N ′][N ′]

E is

a D-algebra. For this purpose, observe that L is an indecomposable C-module which belongs tothe same tube of T where N ′ lies. Indeed, since M(b) �= 0 then HomC(N ′,L) �= 0 and hencethe module L lies in T or in the preinjective component of ΓC . If L is a preinjective module,the Tits form of [L]C is not weakly nonnegative by [47]. Hence N ′ and L belong to the sametube in T . If L is not simple regular, we proceed as in (the dual of) case (II.2) below. So, we


may assume that L is simple regular and therefore N ′ = L. Hence E is a D-extension of the coilalgebra [L]C. Thus E is a D-algebra. The shape of E is depicted below

C

• • w

•b

•w′

In case M[L]B = M , then as in the first case we get that the algebra Λ[M] is a D-algebra,a contradiction. In case, M[L]B �= M then, proceeding as above, E1 = [L′][L]C[N ′] is a con-vex subcategory of A which is a D-algebra. Moreover, L′ = N ′ = L and satisfies that ME1 = M .This implies also that Λ[M] is a D-algebra, as desired.

(ii) Assume N ′ is not isomorphic to Ni for any i = 1, . . . , s. As in the case (α) all modules Ni

and N ′ lie in the same tube of T . Then B[N ′] is a coil algebra, contradicting the maximalityof B .

(II.2) N ′ has regular length r � 2. In that case, N ′ belongs to a tube of rank n − 2 andr = 2. Therefore the algebra C[N ′] is a pg-critical algebra. As above, if N ′ = M , then thepushout Λ[M] = Λ�

CC[M] of Λ and C[M] along C is a D-algebra containing properly Λ,

which contradicts the maximality of Λ. In case N ′ �= M , we get a convex subcategory of A ofthe form E1 = [L]C[N ′] which is a D-algebra and (as it is not difficult to show) ME1 = M .

We consider now the situation where the restriction NC = 0, that is, one of the followingsituations occur:

(a) There is a convex subcategory B1 of B of the form [N1 ⊕ Y ](C ⊕ D) which is a coilcoextension of C of type (ad 1)*, where D is an upper triangular algebra with quiver

a = a1 • · · · ar · · · as · · · at

moreover, Y is the projective–injective D-module and suppN is contained in D. In this caseassume that suppN is the interval [ar , as].

(b) There is a convex subcategory B1 of B of the form [N1]C which is a coil coextensionof C of type (ad 2)* or (ad 3)* and such that suppN ⊂ {w}, where w is the coextension vertexof [N1]C.

We consider these cases with some subcases:(a) Suppose suppN = [ar , as] (with r � 2) lies in the branch of the following coil algebra C′

C

• • · · · • • · · · • · · · •
a ar as at


If r = t , the weak nonnegativity of the Tits form implies that B[N ] is either a coil algebra ora tubular algebra. By hypothesis, B[N ] is a coil algebra, contradicting the maximality of B . Ifr < t , since qA is weakly nonnegative, then either r = s − 1 or r = s. We distinguish two cases.

(a.1) Assume r = s − 1. Then the extension B ′ := C′[N ] has the shape

C

•w

•a1

• · · · • •as−1

•as

•as+1

· · · •at

which is a D-algebra and therefore B[N ] = B �C′ B

′ is also a D-algebra. This implies that Λ[M]is a D-algebra, as we next show, which is a contradiction.

Indeed, if M = N , then as above, Λ[M] = B[M]�B

Λ is a D-algebra. In case M �= N , consider

a vertex b with M(b) �= 0 = N(b) such that the vertices of B[N ] and b form a convex subcate-gory B ′ = [L]B[N ] of A for some indecomposable module L. Then one of the following threesituations occurs.

(i) There is no arrow from any ai to b. Then B ′ has the following shape

C •b

•w

•a1

• · · · • •as−1

•as

•as+1

· · · •at

Then qB ′ is not weakly nonnegative, which is a contradiction. Indeed, observe that the algebra B1obtained from B ′ by deleting the vertices as+1, . . . , at is a pg-critical algebra with a criticalsubcategory C1 containing the vertices as and w. Therefore the extension [L]B1 contains thewild algebra [L]C1 which accepts a vector x with q[L]C1(x) < 0 by [47].

(ii) There is an arrow as−1 → b. Then B ′ has the shape

C

•w

•a1

• · · · •aj

· · · • • •as

•as+1

· · · •at

•b

Observe that since qA is weakly nonnegative then A(aj , b) = 0 for some j − 3 � j � s − 2. Ifj = s − 2, then the algebra [L]B is a coil algebra, contradicting the maximality of B . Hence


j = s − 3. In this case, the algebra B1 = [L]B is a D-algebra such that MB1 = M , thereforeshowing that Λ[M] is a D-algebra.

(iii) There is an arrow as → b. By the weak nonnegativity of qA, the algebra B ′ has the shape

C

•w

•a1

• · · · •as−2

• • •as+1

· · · •at

•b

Let B ′ = [L]B[N ′] with L an indecomposable module. Then B ′ is a D-algebra. In fact, a simpleinspection yields that in this case M[L]B = M . The result follows.

(a.2) Assume r = s. Then the extension B ′ := C′[N ] has the shape

C

•w

•a1

• · · · • •as−1

•as

•as+1

· · · •at

which is a D-coil algebra, contradicting the maximality of B .(b) We may dually consider the situation B1 = C[N1] a coil extension of C of type (ad 2)

or (ad 3) with extension vertex w and N = Pw the projective B1-module at the vertex w. Thecategory HomC[N1](N,−)|C , where C is the coil in ΓC[N1] where N lies, accepts a subposet S

whose Hasse diagram is of the form:

•• •• •

N4 = • • •

We illustrate the construction of S in the case B1 is a coil extension of type (ad 2) where thesupport S(N1) is of the form

Y1 N1 = X0 X1 X2 · · ·

Then the coil C contains a full translation subquiver of the form (see [3,4])


Y1 X1

X0 Pw Z11 X2

X1 Z21 X3

X2 Z31 X4

X3 Z41 X5

X4 Z51 . . .

X5 . . .

. . .

The modules X1, Z21, Z31, Z41 and X5, . . . ,X8 form a poset of type N4 as desired. Recall thatS belongs to the Nazarova’s list and, moreover, to the extension S of S by a maximal point m

corresponds a vector v = w + em, with w indicated in the picture

1

2 2

4 4

4 6 6

such that qS(v) < 0, where q

Sis the poset quadratic form of S (see [65, Chapter 10]). To v

corresponds a vector z = 8eb + ∑i∈S vi dimXi in the Grothendieck group of B1, where Xi is

the indecomposable module in the vertex i in S as subset of ΓC[N1] and b is the extension vertexof B1[N ]. As in [42], we get qB1[N ](z) = q

S(v) < 0, contradicting the fact that qA is weakly

nonnegative.(III) Suppose that 0 �= N ∈ IB . Consider the critical subcategory C of B and the restriction

N ′ = NC . By case (II), we may assume that 0 �= N ′ ∈ I , where I is the preinjective componentof ΓC and let indC = P ∨ I be a splitting of indC. Let C = B0,B1, . . . ,Bs = A be a familyof convex subcategories of A such that, for each 0 � i � s, we have Bi+1 = Bi[Mi] or Bi+1 =[Mi]Bi for some indecomposable Bi -module Mi .

For 0 < i < s − 1, in case that Bi+1 = Bi[Mi], we may suppose that Ni = (Mi)C ∈ I , other-wise cases (I) and (II) yield the result. Dually, in case that Bi+1 = [Mi]Bi , we may suppose thatNi = (Mi)C ∈ P . Therefore the hypotheses of Proposition 5.4 are satisfied and we conclude thatA is a tilted algebra. This is a contradiction which completes the proof of the Main Theorem. �


References

[1] I. Assem, D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras 1: Techniquesof Representation Theory, London Math. Soc. Stud. Texts, vol. 65, Cambridge University Press, Cambridge, 2006.

[2] I. Assem, A. Skowronski, On some classes of simply connected algebras, Proc. Lond. Math. Soc. 56 (1988) 417–450.

[3] I. Assem, A. Skowronski, Multicoil algebras, in: Representation of Algebras, in: CMS Conf. Proc., vol. 14, Amer.Math. Soc., Providence, RI, 1993, pp. 29–68.

[4] I. Assem, A. Skowronski, Coil and multicoil algebras, in: Representation Theory of Algebras and Related Topics,in: CMS Conf. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 1–24.

[5] I. Assem, A. Skowronski, B. Tomé, Coil enlargements of algebras, Tsukuba J. Math. 19 (1995) 453–479.[6] M. Auslander, I. Reiten, S.O. Smalø, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math.,

vol. 36, Cambridge University Press, Cambridge, 1995.[7] R. Bautista, P. Gabriel, A.V. Roiter, L. Salmeron, Representation-finite algebras and multiplicative bases, Invent.

Math. 81 (1985) 217–285.[8] R. Bautista, F. Larrion, L. Salmeron, On simply connected algebras, J. Lond. Math. Soc. 27 (1983) 212–220.[9] K. Bongartz, Treue einfach zusammenhängende Algebren. I, Comment. Math. Helv. 57 (1982) 282–330.

[10] K. Bongartz, Algebras and quadratics forms, J. Lond. Math. Soc. 28 (1983) 461–469.[11] K. Bongartz, Critical simply connected algebras, Manuscripta Math. 46 (1984) 117–136.[12] K. Bongartz, A criterion for finite representation type, Math. Ann. 269 (1984) 1–12.[13] K. Bongartz, P. Gabriel, Covering spaces in representation theory, Invent. Math. 65 (1982) 331–378.[14] K. Bongartz, C.M. Ringel, Representation-finite tree algebras, in: Representation of Algebras, in: Lecture Notes in

Math., vol. 903, Springer, 1981, pp. 39–54.[15] S. Brenner, Decomposition properties of some small diagrams of modules, Sympos. Math. Inst. Naz. Alta Mat. 13

(1974) 127–141.[16] S. Brenner, Quivers with commutativity conditions and some phenomenology of forms, in: Representations of

Algebras, in: Lecture Notes in Math., vol. 488, Springer, 1975, pp. 29–53.[17] O. Bretscher, P. Gabriel, The standard form of a representation-finite algebra, Bull. Soc. Math. France 111 (1983)

21–40.[18] T. Brüstle, On positive roots of pg-critical algebras, Linear Algebra Appl. 365 (2003) 107–114.[19] T. Brüstle, Tame tree algebras, J. Reine Angew. Math. 567 (2004) 51–98.[20] M.C.R. Butler, C.M. Ringel, Auslander–Reiten sequences with few middle terms and applications to string algebras,

Comm. Algebra 15 (1987) 145–179.[21] W.W. Crawley-Boevey, On tame algebras and bocses, Proc. Lond. Math. Soc. 56 (1988) 451–483.[22] V. Dlab, C.M. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 173 (1976).[23] P. Donovan, M.R. Freislich, The Representation Theory of Finite Graphs and Associated Algebras, Carleton Lecture

Notes, vol. 5, Carleton Univ., Ottawa, 1975.[24] P. Dowbor, A. Skowronski, Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62 (1987)

311–337.[25] P. Dräxler, Über einen Zusammenhang zwischen darstellungsendlichen Algebren und geordneten Mengen,

Manuscripta Math. 60 (1988) 349–377.[26] P. Dräxler, Aufrichtige gerichtete Ausnahmealgebren, Bayreuth. Math. Schr. 29 (1989).[27] P. Dräxler, Completely separating algebras, J. Algebra 165 (1994) 550–565.[28] P. Dräxler, R. Nörenberg, Thin start modules and representation type, in: Representations of Algebras, in: CMS

Conf. Proc., vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 149–163.[29] P. Dräxler, A. Skowronski, Biextensions by indecomposable modules of derived regular length 2, Compos. Math.

117 (1999) 205–221.[30] Yu.A. Drozd, Tame and wild matrix problems, in: Representation Theory II, in: Lecture Notes in Math., vol. 832,

Springer, 1980, pp. 242–258.[31] P. Gabriel, Unzerlegbare Darsellungen I, Manuscripta Math. 6 (1972) 71–103.[32] P. Gabriel, L.A. Nazarova, A.V. Roiter, V.V. Sergeichuk, D. Vossieck, Tame and wild subspace problems, Ukrainian

J. Math. 45 (1993) 335–372.[33] C. Geiss, On degenerations of tame and wild algebras, Arch. Math. (Basel) 64 (1995) 11–16.[34] D. Happel, D. Vossieck, Minimal algebras of infinite representation type with preprojective component, Manuscripta

Math. 42 (1983) 224–243.


[35] S. Kasjan, Tame strongly simply connected algebras form an open scheme, J. Pure Appl. Algebra 208 (2007) 435–443.

[36] O. Kerner, Tilting wild algebras, J. Lond. Math. Soc. 39 (1989) 29–47.[37] M. Kleiner, Partially ordered sets of finite type, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 28

(1972) 32–41 (in Russian).[38] H. Kraft, Geometric methods in representation theory, in: Representations of Algebras, in: Lecture Notes in Math.,

vol. 944, Springer, Berlin, New York, 1982, pp. 180–258.[39] M. Lersch, Minimal wilde Algebren, Diplomarbeit, Düsseldorf, 1987.[40] P. Malicki, A. Skowronski, Algebras with separating almost cyclic coherent Auslander–Reiten components, J. Al-

gebra 291 (2005) 208–237.[41] P. Malicki, A. Skowronski, B. Tomé, Indecomposable modules in coils, Colloq. Math. 93 (2002) 67–130.[42] N. Marmaridis, J.A. de la Peña, Quadratic forms and preinjective modules, J. Algebra 134 (1990) 326–343.[43] L.A. Nazarova, Representations of quivers of infinite type, Izv. Akad. Nauk SSSR 37 (1973) 752–791 (in Russian).[44] L.A. Nazarova, Partially ordered sets of infinite type, Izv. Akad. Nauk SSSR 39 (1975) 963–991 (in Russian).[45] R. Nörenberg, A. Skowronski, Tame minimal non-polynomial growth strongly simply connected algebras, in: Rep-

resentation Theory of Algebras, in: CMS Conf. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 1996, pp. 519–538.[46] R. Nörenberg, A. Skowronski, Tame minimal non-polynomial growth simply connected algebras, Colloq. Math. 73

(1997) 301–330.[47] J.A. de la Peña, On the representation type of one-point extensions of tame concealed algebras, Manuscripta

Math. 61 (1988) 183–194.[48] J.A. de la Peña, Algebras with hypercritical Tits form, in: Topics in Algebra, in: Banach Center Publ., vol. 26, Part 1,

PWN, Warsaw, 1990, pp. 353–359.[49] J.A. de la Peña, On the dimension of module varieties of tame and wild algebras, Comm. Algebra 19 (1991) 1795–

1807.[50] J.A. de la Peña, Tame algebras with sincere directing modules, J. Algebra 161 (1993) 171–185.[51] J.A. de la Peña, The families of two-parametric tame algebras with sincere directing modules, in: Representations

of Algebras, in: CMS Conf. Proc., vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 361–392.[52] J.A. de la Peña, On the corank of the Tits form of a tame algebra, J. Pure Appl. Algebra 107 (1996) 159–183.[53] J.A. de la Peña, The Tits form of a tame algebra, in: Representation Theory of Algebras and Related Topics, in:

CMS Conf. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 159–183.[54] J.A. de la Peña, A. Skowronski, Geometric and homological characterizations of polynomial growth strongly simply

connected algebras, Invent. Math. 126 (1996) 287–296.[55] J.A. de la Peña, A. Skowronski, Forbidden subcategories of non-polynomial growth tame simply connected algebras,

Canad. J. Math. 48 (1996) 1018–1043.[56] J.A. de la Peña, A. Skowronski, Substructures of non-polynomial growth algebras with weakly non-negative Tits

form, in: Algebras and Modules II, in: CMS Conf. Proc., vol. 24, Amer. Math. Soc., Providence, RI, 1998, pp. 415–431.

[57] J.A. de la Peña, A. Skowronski, The Tits and Euler forms of a tame algebra, Math. Ann. 315 (1999) 37–59.[58] J.A. de la Peña, A. Skowronski, Substructures of algebras with weakly non-negative Tits forms, Extracta Math. 22

(2007) 67–81.[59] J.A. de la Peña, A. Skowronski, Algebras with cycle-finite Galois coverings, Trans. Amer. Math. Soc., in press.[60] J.A. de la Peña, B. Tomé, Iterated tubular algebras, J. Pure Appl. Algebra 64 (1990) 303–314.[61] I. Reiten, A. Skowronski, Characterizations of algebras with small homological dimensions, Adv. Math. 179 (2003)

122–154.[62] C.M. Ringel, Tame algebras, in: Representation Theory I, in: Lecture Notes in Math., vol. 831, Springer, 1980,

pp. 137–287.[63] C.M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math., vol. 1099, Springer-Verlag,

Berlin, 1984.[64] A. Rogat, T. Tesche, The Gabriel quivers of the sincere simply connected algebras, Preprint 93-005, SFB 353

(Bielefeld).[65] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra Logic Appl.

Ser., vol. 4, Gordon Breach Sci. Publ., Montreux, 1992.[66] D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras 2: Tubes and Concealed

Algebras of Euclidean Type, London Math. Soc. Stud. Texts, vol. 71, Cambridge University Press, Cambridge, 2007.[67] D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras 3: Representation-

Infinite Tilted Algebras, London Math. Soc. Stud. Texts, vol. 72, Cambridge University Press, Cambridge, 2007.


[68] A. Skowronski, Algebras of polynomial growth, in: Topics in Algebra, in: Banach Center Publ., vol. 26, Part 1,PWN, Warsaw, 1990, pp. 535–568.

[69] A. Skowronski, Simply connected algebras and Hochschild cohomologies, in: Representation of Algebras, in: CMSConf. Proc., vol. 14, Amer. Math. Soc., Providence, RI, 1993, pp. 431–447.

[70] A. Skowronski, Simply connected algebras of polynomial growth, Compos. Math. 109 (1997) 99–133.[71] A. Skowronski, Tame algebras with strongly simply connected Galois coverings, Colloq. Math. 72 (1997) 335–351.[72] A. Skowronski, Tame quasi-tilted algebras, J. Algebra 203 (1998) 470–490.[73] A. Skowronski, J. Waschbüsch, Representation-finite biserial algebras, J. Reine Angew. Math. 345 (1983) 172–181.[74] L. Unger, The concealed algebras of the minimal wild, hereditary algebras, Bayreuth. Math. Schr. 31 (1990) 145–

154.[75] B. Wald, J. Waschbüsch, Tame biserial algebras, J. Algebra 95 (1985) 480–500.[76] J. Wittman, Verkleidete zahme und minimal wilde Algebren, Diplomarbeit, Bayreuth, 1990.[77] M.V. Zeldich, A criterion for weakly positive quadratic forms, in: Linear Algebra and Representation Theory, Inst.

Math. Acad. Sci. Ukrainian SSR, Kiev, 1983, pp. 125–137.

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Tame algebras and Tits quadratic forms